iqst.ca · Abstract Nonlinear effects are ubiquitous in nature. Many interesting phenomena are described by differ-ential equations that are nonlinear. Even richer dynamics can be

UNIVERSITY OF CALGARY

Nonlinear dynamics of mathematical models and proposed implementations in ultracold atoms

by

Hon Wai Lau

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAM IN PHYSICS AND ASTRONOMY

CALGARY, ALBERTA

JULY, 2017

c© Hon Wai Lau 2017

Abstract

Nonlinear effects are ubiquitous in nature. Many interesting phenomena are described by differ-

ential equations that are nonlinear. Even richer dynamics can be observed with additional long-

range spatial coupling. For example, an interesting type of pattern discovered recently can form

in systems with nonlocal coupling. The pattern, called chimera states, is composed of both phase

coherent and incoherent regions coexisting in the same system. Through numerical studies of os-

cillatory media with nonlocal diffusive coupling, I show here for the first time that stable chimera

knot structures can exist in 3D. Knots were not previously known to be stable in oscillatory media,

nor were such non-trivial chimera patterns known to exist in 3D.

To realize different nonlinear phenomena in a controlled way in experiments, a flexible physical

system is required. Ultracold atomic systems, specifically, Bose-Einstein condensates (BECs),

are good candidates because of the high controllability of almost all parameters, including the

nonlinearity, in real time. Hence, experimental studies can be carried out for a variety physical

systems, including many classical and quantum field equations. In particular, in this thesis I study

a setup of BECs with a third-order Kerr nonlinearity to generate Schrodinger cat states, which have

applications in quantum metrology. I showed that cat states involving hundreds of atoms should be

realizable in BECs. This requires careful optimization of the experimental parameters and analysis

of the atom loss.

Inspired by the previous two projects, it is an interesting question if chimera states can exist in

BECs. By analyzing the underlying mechanism of the effective nonlocal diffusive coupling, I es-

tablish here a new analogous mechanism to achieve mediated nonlocal spatial hopping for particles

in BECs with two interconvertible states. By adiabatically eliminating the fast mediating channel, I

obtain the mean-field of Bose-Hubbard model with fully tunable hopping strength, hopping range,

and nonlinearity. This is the first known conservative system exhibiting chimera patterns. More

importantly, I show that the model should be implementable in BECs with current technology.

ii

List of all published papers during PhD

1. Hon Wai Lau, Jorn Davidsen, Christoph Simon, “Matter-wave mediated hopping in

ultracold atoms: Chimera patterns in conservative systems”, To be submitted.

2. Hon Wai Lau, Jorn Davidsen, “Linked and knotted chimera filaments in oscillatory

systems”, Phys. Rev. E 94, 010204(R) (2016). [arXiv:1509.02774]

3. Mohammadsadegh Khazali, Hon Wai Lau, Adam Humeniuk, Christoph Simon,

“Large Energy Superpositions via Rydberg Dressing”, Phys. Rev. A 94, 023408

(2016). [arXiv:1509.01303]

4. Tian Wang, Hon Wai Lau, Hamidreza Kaviani, Roohollah Ghobadi, Christoph Si-

mon, “Strong micro-macro entanglement from a weak cross-Kerr nonlinearity”,

Phys. Rev. A 92, 012316 (2015). [arXiv:1412.3090]

5. Hon Wai Lau, Zachary Dutton, Tian Wang, Christoph Simon, “Proposal for the

Creation and Optical Detection of Spin Cat States in Bose-Einstein Condensates”,

Phys. Rev. Lett. 113, 090401 (2014). [arXiv:1404.1394]

6. Hon Wai Lau, Peter Grassberger, “Information theoretic aspects of the two-dimensional

Ising model”, Phys. Rev. E 87, 022128 (2013). [arXiv:1210.5707]

7. Hon Wai Lau, Maya Paczuski, Peter Grassberger, “Agglomerative percolation on

bipartite networks: Nonuniversal behavior due to spontaneous symmetry breaking

at the percolation threshold”, Phys. Rev. E 86, 011118 (2012). [arXiv:1204.1329]

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Acknowledgements

First of all, I would like to express my deep gratitude to Christoph Simon, my PhD supervisor,

for his patience, encouragement, and advises, during my PhD study. He is able to give excellent

guidance even if the projects I worked on is outside of his expertise. Research in our group is

enjoyable, since he is actively improving the environment and reducing the stress of students. I am

always impressed by his ability to find out the key points in new papers only in couple of seconds

and to simplify complicated problems.

I especially thanks for the complexity science group at University of Calgary during my early

PhD. The group was great and I enjoyed the occasional brainstorming with Peter Grassberger. I

hoped that I could have worked with him for longer time. I also thanks for the guidance from

Jorn Davidsen, in which the works with him formed part of this thesis. Thanks Golnoosh Bizhani,

Aicko Yves Schumann, Chad Gu, and Arsalan Sattari for the advises during my hard time.

It was a great pleasure to work with my group members. I have had lots of long and insight-

ful discussions, not only limited to academics, with Stephen Wein, Tian Wang, Sourabh Kumar,

Sandeep Goyal, Mohammadsadegh Khazali, James Moncreiff, Sumit Goswami, Farid Ghobadi,

and Khabat Heshami. Thanks for my friends Wei-wei Zhang, Akihiko Fujii, Jiawei Ji, Yadong

Wu, Adarsh Prasad, Ish Dhand, and many others and people I met during my study here. All of

them make the life here more fun.

I would also like to thank for the collaborators and the academic help from Farokh Mivehvar,

Zachary Dutton, Lindsay J. LeBlanc, and the people that I get help through email exchanges includ-

ing someone who I never met before. Thanks David Hobill, Michael K. Y. Wong, Alex Lavovsky,

Barry Sanders, David Feder, and Maya Paczuski for sharing their knowledge, as well as offering

academic and career advises.

A special thank goes to Kwok Yip Szeto, my undergraduate and MPhil research supervisor, for

leading me into the academic world.

iv

I would like to express my thanks to all my old friends in Hong Kong and scattered over the

world. Specifically, the friends going through the academic life and exchanging experiences with

me: Yun kuen Cheung, Cheung Chan, Alan Fung, Alan Tam, Lokman Tsui, Tin Yau Pang. They

are like my companions in this long academic journey.

Last but not least, I would like to thank my parents, sister, and brother for their supports, which

allows me to pursue my goal freely.

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiList of all published papers during PhD . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 A new type of pattern - chimera states . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Quantum effects in macroscopic systems - Schrodinger cat states . . . . . . . . . . 81.3 Simulating physical systems - Bose Einstein condensates . . . . . . . . . . . . . . 111.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Oscillators and phase space dynamics . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Simple harmonic oscillators and nonlinear oscillators . . . . . . . . . . . . 182.1.2 Quantum oscillators, Kerr nonlinearity and cat states . . . . . . . . . . . . 23

2.2 Field equations and nonlocal coupling . . . . . . . . . . . . . . . . . . . . . . . . 272.2.1 Local coupling and complex Ginzburg-Landau equation . . . . . . . . . . 272.2.2 Nonlocal diffusive coupling . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.3 Nonlocal hopping with mediating channel . . . . . . . . . . . . . . . . . . 31

2.3 Bose-Einstein Condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.1 Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.2 Kerr nonlinearity in BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.3 Kerr nonlinearity in two-component BEC . . . . . . . . . . . . . . . . . . 352.3.4 Mean-field equation of two-component BEC . . . . . . . . . . . . . . . . 37

3 Linked and knotted chimera filaments in oscillatory systems . . . . . . . . . . . . 393.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Phase oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 Existence of knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.6 Dependence on R, L, and geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 463.7 Robustness with respect to noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.8 Dependence on spatial kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.9 Beyond phase oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.10 Complex oscillatory systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.11 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.12 Appendix A: Topological structures . . . . . . . . . . . . . . . . . . . . . . . . . 523.13 Appendix B: Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.14 Appendix C: Creating chimera knots . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.14.1 Random initial condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.14.2 Algorithm to create rings and Hopf links . . . . . . . . . . . . . . . . . . 543.14.3 Reconnecting chimera filaments using random patches . . . . . . . . . . . 58

3.15 Appendix D: Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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3.15.1 Instability of a single ring . . . . . . . . . . . . . . . . . . . . . . . . . . 613.15.2 Instability of knots for R = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 623.15.3 Filament instability at α0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.15.4 Instabilities from finite size effects . . . . . . . . . . . . . . . . . . . . . . 62

3.16 Appendix E: Spatial kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.17 Appendix F: Other oscillatory models . . . . . . . . . . . . . . . . . . . . . . . . 65

3.17.1 Non-Local Complex Ginzburg-Landau equation (CGLE) . . . . . . . . . . 653.17.2 CGLE: Minimum separation & spontaneous fluctuations . . . . . . . . . . 663.17.3 Non-Local Rossler model . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Proposal for the Creation and Optical Detection of Spin Cat States in Bose-EinsteinCondensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Spin cat states creation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.4 Calculating energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5 Cat creation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.6 Atom loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.7 Detection scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.9 Appendix A: Properties of two-component BEC . . . . . . . . . . . . . . . . . . . 844.10 Appendix B: Ground state energy from first order perturbation theory . . . . . . . 884.11 Appendix C: Effects of higher-order nonlinearities . . . . . . . . . . . . . . . . . 904.12 Appendix D: Phase separated regime and non-phase separated regime . . . . . . . 924.13 Appendix E: Atom loss rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.14 Appendix F: Readout loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.15 Appendix G: Allowable uncertainty in atom number . . . . . . . . . . . . . . . . . 964.16 Appendix H: Atom loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.17 Appendix I: Comparison with photon-photon gate proposal . . . . . . . . . . . . . 1015 Matter-wave mediated hopping in ultracold atoms: Chimera patterns in conserva-

tive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3 Nonlocal hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.4 Mediating mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.5 Implementation in ultracold atomic systems . . . . . . . . . . . . . . . . . . . . . 1085.6 Dynamics and chimera patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.7 Experimental settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.8 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.9 Appendix A: Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.10 Appendix B: Ultracold atom with periodic lattice . . . . . . . . . . . . . . . . . . 1155.11 Appendix C: Hopping Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.12 Appendix D: Chimera patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.13 Appendix E: Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.13.1 Split method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.13.2 Time split method for the one-component GPE . . . . . . . . . . . . . . . 128

vii

5.13.3 Time split method for the two-component GPE . . . . . . . . . . . . . . . 1296 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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List of Symbols, Abbreviations and Nomenclature

Symbol Definition

BEC Bose-Einstein Condensate

GPE Gross-Pitaevskii equation

CLGE Complex Ginzburg-Landau equation

IC Initial condition

BC Boundary condition

1D One dimension

2D Two dimension

3D Three dimension

SDS Synchronization defect sheets

TFA Thomas-Fermi approximation

CSS Coherent spin state

BHM Bose-Hubbard model

NLHM nonlocal hopping model

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Chapter 1

Introduction

Many interesting dynamics and physical phenomena observed in nature can only be modeled by

nonlinear differential equations. These phenomena include chaos, solitons, and many patterns that

are unique to nonlinear systems. One of the most famous nonlinear differential equations yielding

a variety of different patterns is the complex Ginzburg-Landau equation (CGLE) [1, 2], which

describes many physical systems phenomenologically, such as superconductivity and nonlinear

waves. The CGLE is famous because it is the normal form of any system close to a Hopf bifurcation

- a critical point where a stationary system begins to oscillate [2, 3].

Chimera states are a particular type of pattern that have been recently discovered [4, 5], which

are states that contain oscillators synchronized in some region, but unsynchronized in another

region (see Fig. 1.3). This happens for oscillators that are completely identical. It is understood

that the nonlocal diffusive coupling plays a key role in the formation of most chimera patterns.

In particular, many chimera patterns can be observed in the nonlocal CGLE in 1D and 2D [6, 7].

In this thesis, I will present a new chimera pattern called chimera knots in 3D that have never

been seen before [8]. It is a pattern with synchronized regions everywhere except around a knot

structure in 3D. This is an important discovery because no stable knots were known to exist with

local coupling in CGLE, or more generally, oscillatory media.

Another famous and related nonlinear differential equation is the Gross-Pitaevskii equation

(GPE) [9], which was derived as a mean-field description of Bose-Einstein condensates (BECs).

A BEC is a state with all bosons occupying the same single particle state [10]. Hence, in the

limit of large numbers of particles, a simple description is possible by taking the mean of the

corresponding quantum field equation . The GPE is the Schrodinger equation but with an extra

nonlinearity originating from two-particle collisions, and it is identical to the CGLE in certain

1

parameter regimes.

In quantum systems, the nonlinearity from two-particle collisions can be used to generate

Schrodinger cat states, which are superpositions of two macroscopically distinct states. The cat

states in BECs can involve many atoms and are very sensitive to particle loss. Even the loss of a

single particle can cause significant decoherence and destroy the cat states. Therefore, analyzing

the effects of particle loss is important when making practical scheme proposals. Large cat states

have applications from precise quantum metrology that can surpass the standard quantum limit,

to detect the hypothetical energy collapse model [11]. In this thesis, I will propose and analyze a

scheme using two-component BECs to generate cat states [12].

The differential equations CGLE and GPE are very similar to each other. For example, both

of them have a third-order nonlinearity, are equivalent in some regime, and show similar patterns.

This close relationship between CGLE and GPE suggests that chimera states may also exist in

BECs. As mentioned, the nonlocal diffusive coupling in the CGLE is important for chimera states,

so it is reasonable to speculate that there may be an analogue for the GPE too. After studying

the underlying mediating mechanism of the nonlocal diffusive coupling, I have found such an

analogue to be the mediated nonlocal hopping. The idea is similar to the typical mediating mecha-

nisms for particle-particle interactions such as Coulomb’s law: The complete description typically

involves the consideration of the particle-field-particle interaction. However, if the field is orders

of magnitude faster than the interesting dynamics of the particles, then the field can be eliminated

adiabatically, resulting in an effective, simple, and instantaneous nonlocal description. By using

two-component BECs with one of the components treated as a mediating channel with matter-

wave as a mediating field, I show that an effective mediated nonlocal hopping can appear. This

nonlocal hopping model (NLHM) allows experimentally tunable hopping range, hopping strength

and, nonlinearity. Since ultracold atomic BECs are highly controllable, I have also found an exper-

imentally realistic parameter regime in which my proposed mechanism should work, and where

chimera states should be observable. This discovery will allow the study of chimera states in BECs

2

(a) (b) (c)

Figure 1.1: Spatio-temporal patterns in 1D. The horizontal axis is space x, and the vertical axis istime t. Values increases from blue to red. (a,b) Amplitude A(x, t) of two systems. (c) Phase θ(x, t)of (b).

and conservative systems. The quantization of this NLHM is the Bose-Hubbard model [13] with

nonlocal hopping, and there are likely interesting physics and quantum phases to be discovered.

In the remainder of this chapter, I will introduce the three main topics involved in this thesis:

chimera states, cat states, and Bose-Einstein condensates. The theoretical background required to

understand my results will be presented in the following chapter.

1.1 A new type of pattern - chimera states

Various interesting structures exist everywhere in nature, and easily recognizable structures are of-

ten referred to as patterns [3, 15, 16, 17]. A pattern may be considered as a high-level macroscopic

description of the structures in different subparts of a given system. Patterns are usually visually

distinct such as the spatio-temporal patterns in Fig. 1.1, or snapshots of spatial patterns in Fig.

1.2. These distinct features suggest that mathematical modeling is possible. For example, if the

local dynamics of every spatial location r are oscillators, then the system may be described by two

3

Figure 1.2: Snapshots of the patterns of in 2D. Amplitude A(r, t) and phase θ(r, t) for (a,e) asingle spiral, (b,f) multiple spirals, and (c,g) turbulence-like pattern. (d) Plane wave. (h) A linedefect from the center to the left edge in Rossler model [14].

dynamical variables, the amplitude field A(r, t) and the phase field θ(r, t). A new type of pattern

called chimera states has been discovered recently, as shown Fig. 1.3a-e. The visual feature of

this pattern is the coexistence of phase coherence in one region and phase incoherence in another

region.

This chimera pattern is surprising because it can exist in a system with identical oscillators and

identical coupling between oscillators. Before the discovery in 2002, for a long time, a network of

identical oscillators was believed to be relatively boring with only two possibilities: Fully coherent

or incoherent [5]. This viewpoint changed in 2002 when Yoshiki Kuramoto and his collaborator

Dorjsuren Battogtokh were studying a ring of identically and nonlocally coupled phase oscillators

[4]. They discovered that, for certain initial conditions, some of the oscillators can synchronize

while the remaining oscillators are incoherent. This happens even in systems with translational

and rotational symmetry in 2D studied in the follow-up studies [7, 18, 19]. The patterns include

coherent and incoherent spots, stripes and spirals with randomized cores as shown in Fig. 1.3. The

term chimera state was coined by Steve Strogatz [20] in 2004 because of the similarity with the

Greek mythological creature composed of different animals.

4

(a) α = 1.35,R = 65 (b) α = 1.55,R = 85 (c) α = 1.5,R = 90 (d) α = 0.7,R = 12

(e) α = 0.85,R = 12 (f) α = 1.2,R = 12 (g) α = 1.55,R = 12 (h) α = 1.7,R = 12

Figure 1.3: Snapshots of patterns observed in the nonlocal Kuramoto model (see Chapter 3).The plots are the phase θ(r, t) of oscillators. (a) Incoherent spot. (b) Coherent spot. (c) Chimerastrip. (d) Incoherent core with a spiral. (e) Multiple incoherent cores. (f) Irregular pattern with theexpansion of cores. (g) Near-random pattern. (h) Completing plane wave. Initial condition for allpatterns are random phase, except (d) which starts from a spiral. Each direction has length L = 200oscillators.

Only a decade later, in 2012, the existence of chimera states in experimental systems was con-

firmed in two demonstrations. The first one used photosensitive chemical oscillators with the light

feedback coupling in a two-cluster setup [21], and later in a 1D ring [22]. The second experiment

used a coupled map lattice with coupling through camera detections and light feedback [23]. The

criticism of computer-controlled coupling was addressed by a third experiment using pure me-

chanical oscillators in 2013 [24]. The success of these experiments raises the interest of explaining

physical phenomena in systems that resemble oscillator dynamics such as brains, hearts, and power

grids [5].

As mentioned above, a chimera state is defined as a state with subpopulations that are mutually

synchronized and the remaining populations unsynchronized. The term synchronization, as defined

in [25], is the adjustment of rhythms of oscillating objects due to their weak interaction. A simple

5

harmonic oscillator is an example of an oscillating object. Most oscillators with multiple oscillating

cycles need an input of energy to be self-sustained such as clocks, pendulum, lasers, chemical

oscillation, pacemakers, and neuron activity. For these systems, a natural frequency ω0 and phase θ

can be defined. The coupling between oscillators change the rhythms, or the oscillating frequency

and phase dynamics, of the oscillators. The synchronization of oscillators occurs when different

natural frequencies that become locked to a common frequency due to weak coupling [26]. The

simplest mathematical modeling for this phenomenon is the Kuramoto model [27].

The fact that coherent and incoherent regions can coexist in systems with completely identical

oscillators is a prime example of the spontaneous breaking of synchrony. It is worth mentioning

that with specially selected natural frequencies and couplings, patterns similar to chimera states

may be created [5], which may not be considered as chimera states. Chimera states can appear

as spatio-temporal patterns in 1D, 2D, two-cluster, and complex networks as listed in Ref. [5].

Numerical simulations suggest that most chimera states are stable, as well as robust against noises

and perturbations. A few chimera states in 1D are known to be transient states with a long lifetime

and become stable in the thermodynamic limit [28]. There are many classifications of chimera

states. For example, stationary chimera patterns have a stationary boundary between coherence

and incoherence regions as shown in Fig. 1.3(a-c), but still show phase randomness in time. In this

case, an ansatz may be used to simplify the analysis of the stability [29, 30].

Oscillatory media are continuous media where each spatial point can be treated as an oscillator

locally [16]. While in general continuous media u(r, t) with space r, time t, and field u continuum

can be described by partial differential equations. Therefore, synchronization can exist in oscilla-

tory media because oscillator dynamics exist locally and are coupled through, say, diffusion of the

form ∇2u. One of the most-studied and well-known equations for oscillatory media is the complex

Ginzburg-Landau equation (CGLE) [1]. It is the normal form of all oscillatory media close to a

supercritical Hopf bifurcation [17]. A Hopf bifurcation happens when tuning a control parameter

causes a stable point in phase space to become a stable limit cycle oscillation.

6

The developments of CGLE can be traced back to the early work of Lev Davidovich Landau

on phase transitions in 1937 [31, 32]. Later in 1950, together with Vitaly Lazarevich Ginzburg,

they postulated the Ginzburg-Landau theory as a phenomenological description of superconduc-

tors [33]. Similar equations were eventually found to provide good descriptions of diverse phe-

nomena including nonlinear waves, Rayleigh-Benard convection, and Bose-Einstein condensates.

An example of oscillatory media is reaction-diffusion systems where the local reactions behave

like self-sustained oscillators and are coupled through spatial diffusion. Experimentally, it can be

realized by chemical oscillations such as the Belousov-Zhabotinsky chemical reaction [2].

Nonlocal coupling plays a key role in most formations of spatial patterns of chimera states. As

shown in the original works in 1D and 2D [4, 7], chimera states can be observed in the nonlocal

CGLE where the typical diffusion term is replaced by a nonlocally coupled term, and later in the

simplified nonlocal Kuramoto model [18]. A study of chimera states in 3D only happened recently

[34]. In 3D, the point-like phase singularity in 2D at the center of Fig. 1.2a will become a line-like

structure, often called a filament. The spiral wave in 2D becomes a scroll wave in 3D. Instead of a

straight filament, filaments can also be closed like a ring. The stability of straight filaments, rings,

and twisted filaments have been studied [35, 36]. However, no stable knots and links such as Hopf

links (two intertwined rings) and trefoils have been observed. In contrast, stable knots and links

exist in excitable media [37, 38]. The local dynamics of excitable media is normally in a stable

non-oscillating state, and will only go through an oscillating cycle when the perturbation is large

enough. Excitable media are very similar to oscillatory media and show many similar patterns,

so it is surprising that knot structures are not stable in oscillatory media. On the other hand, the

incoherent core in 2D in Fig. 1.3d should become a tube-like structure filled with incoherent

oscillators in 3D. This suggests there are structures combining line-like topological structures with

chimera structures in 3D, as studied in [34]. It is reasonable to expect that stable knots may also

exist. The existence of stable chimera knots in oscillatory media with nonlocal coupling is the

theme of Chapter 3.

7

(a) A coherent state (b) A cat state

Figure 1.4: Quasi-probability distribution of (a) a coherent state, (b) and a cat state, which is thesuperposition of two coherent states. The negative values in the fringes indicate the non-classicalityof the state.

As a new research field, there are still many open questions about the nature of chimera states

as listed in a review paper [5] such as the stability criteria, the necessary conditions for chimera

states, and their relationship with resonance. All chimera states studied until now occur in driven-

dissipative systems that are out of equilibrium. In Chapter 5, I will present the existence of chimera

states in conservative Hamiltonian systems. Moreover, there is evidence that chimera states may

also exist in BECs with a mediated nonlocal hopping, which is an analogue of the nonlocal cou-

pling in the CGLE.

1.2 Quantum effects in macroscopic systems - Schrodinger cat states

One of the most striking aspects of quantum mechanics is the superposition and entanglement of

particles, which have no counterpart in classical systems. In particular, quantum nonlocality due

to entanglement between particles has been conclusively proven in a series of recent experiments.

Loophole free tests of the violation of Bell’s inequality were finally performed by Ronald Hanson’s

group [39], shortly followed by Anton Zeilinger’s group [40] and Lynden Shalm’s group [41] in

2015. There is little doubt that quantum mechanics is the correct description of reality at the

8

microscopic level. However, it is still an open question if quantum mechanics is correct on all

scales. After all, the current form of quantum mechanics is incompatible with general relativity.

The mathematical description of quantum systems has various interpretations that are not in-

tuitive. To illustrate the counter-intuitive nature of applying quantum mechanics to macroscopic

objects, in 1935, Schrodinger proposed a thought experiment now known as Schrodinger’s cat

[42]. In quantum mechanics, a quantum system can be in a superposition of two states, which

can interact differently with a macroscopic object, say, a cat. Suppose the decay of a radioactive

atom triggers a mechanism to kill the cat, while the non-decayed atom will do nothing. If quantum

mechanics works on all scales, then the macroscopic cat will become a superposition of being dead

and alive simultaneously, with the decayed and non-decayed atom respectively. However, when

the chamber is opened, only one of the results can be observed, either an alive cat or a dead cat,

but not both. Such superposition can happen between all kinds of objects, but we never observe

them in our daily life. There are two main reasons for assuming quantum physics is universal.

Firstly, decoherence of quantum systems is effectively a measurement which forces particles to

follow classical mechanics [43, 44]. Hence, superpositions can be destroyed by decoherence due

to interactions with the environment, such as radiation and collision with air molecules, so a large

superposition will decay much faster [45]. Secondly, the required measurement precision grows as

the size of the system grows in general [46], so it is extremely difficult to measure large systems.

Hence, a carefully designed experiment with very high sensitivity is required for measuring any

macroscopic, or even mesoscopic, quantum superposition.

A cat state is defined as a superposition of two macroscopically distinct quasi-classical states.

However, macroscopicity of a state has no unique definition. It can refer to a large spatial extent, a

large mass, or a large number of particles involved. One definition of cat states is the superposition

of two, or more, coherent states in phase space (see Fig. 1.4b), where the coherent states are often

considered to be the most classical particle-like states [47, 48, 49] (see Fig. 1.4a). This definition

is commonly used in quantum optics, and the size of a cat depends on the number of particles. A

9

good quantification of macroscopicity for this definition is provided by Lee and Jeong [50] which

measures the effective size and coherence at the same time. The size of a cat state can be as large

as one hundred particles as achieved in a recent experiment using microwave photons [51].

Applications of cat states include quantum computation [52, 53], which treats a coherent state

as a qubit so that a cat state corresponds to a qubit with a superposition of zero and one. In addition,

cat states are useful in high-precision quantum metrology to bypass the standard quantum limit.

For example, cat states in phase space can reduce the shot-noise of measurements. Cat states of

massive objects can be used to improve the sensitivity of detection of the gravitational waves,

which were recently experimentally confirmed [54]. Energy cat states and position cat states may

be used to detect hypothetical collapse models [55].

Generation of cat states is no easy task. There are a few conceptually different schemes to

generate cat states in phase space. One of the methods is based on bifurcation [56]. In classical

systems, a particle located at an unstable point will go to one of the stable points upon a small

perturbation. However, a corresponding quantum state that starts at the same unstable point will

spread out due to quantum uncertainty, which will result in a superposition state at both stable

points. A similar method allows cat states to be created in open systems [57, 58], which requires

squeezing and two-particle loss to create an unstable point at the origin. A pure quantum mechan-

ical approach was proposed by Yurke and Stoler [59, 60] using the third-order Kerr nonlinearity.

Using this method, starting from a coherent state with the nonlinearity, a cat state will appear at a

certain time because of the phase matching of all number states (see Chapter 2.1). After exactly

the same amount of time, it will evolve back to the coherent state because of the recurrence. The

nonlinearity in BECs takes a similar form of Kerr nonlinearity (see Chapter 2.3). Hence, with a

special setting, atomic cat states involving hundreds of atoms can be created. The detailed analysis

of my proposal will be presented in Chapter 4.

10

Figure 1.5: Velocity distribution of the Rubidium atoms undergoing a Bose-Einstein condensation.The temperature T decreases from left to right, T Tc, T < Tc, and T Tc, where Tc is thetransition temperature. Thermal excitations disappear when the temperature is sufficiently low, soall particles are in the ground state with minimum uncertainty,which is displayed as a sharp peak.(E. Cornell [61]/ Creative Commons)

1.3 Simulating physical systems - Bose Einstein condensates

The experimental realization of Bose-Einstein condensation in ultracold atomic gases in 1995 has

marked a new era of physics and opened up a whole new exciting field that continues to thrive.

After the first realization, BECs quickly attracted lots of attention and were the subject of an ex-

plosion of research, with review articles on different aspects almost every year [9, 62, 63, 64, 65,

66, 67, 68, 69, 70, 71, 72, 73, 74], ranging from experimental techniques to the theory and applica-

tions of BECs. Now, after two decades of development, the relevant experimental techniques have

become mature and are treated as basic tools for atomic physicists to study fundamental physics

and simulate other physical systems.

The theoretical study of BECs dates back to 1924, when Satyendra Nath Bose re-derived the

statistical distribution of photons in the black-body radiation by using the correct way of counting

states of identical and indistinguishable particles, now called bosons. After the paper was rejected

once, Bose sent it to Albert Einstein who agreed with the idea and helped with translating it into

11

German for publication [75]. Extending Bose’s work in the following year, Einstein obtained the

quantum statistical distribution for the non-interacting ideal gas [76], now known as BEC. This

requires cooling the weakly interacting atoms to a temperature a billion times lower than room

temperature and no technique for achieving it existed at the time. The invention of powerful

laser cooling techniques for alkali atoms in the 1970s eventually made it possible to bring the

temperature down to the order of 100µK. The barrier of the remaining two orders of magnitude

was eliminated by further evaporative cooling that let the high energy particles leave the system.

Combining these techniques finally led to the groundbreaking observation of BEC (see Fig. 1.5),

over 70 years after its first prediction [77, 78, 79]. The Nobel Prize in Physics was awarded to Eric

Cornell, Carl Wieman, and Wolfgang Ketterle in 2001 for this achievement [80, 81].

A Bose-Einstein condensate is a state of matter with a macroscopic number of identical bosons

occupying the same single-particle state. The phase transition towards a BEC happens when the

system is cooled below a critical temperature Tc such that the temperature-dependent de Broglie

wavelength becomes larger than the inter-particle separation. At such a low temperature, a macro-

scopic wavefunction forms due to the overlap of the individual wave packets, such that the bosons

lose their distinguishability based on position. In the condensate, every atom behaves exactly

the same. Hence, instead of describing the system by a quantum field ψ(r), a simple mean-

field treatment ψ(r)→ 〈ψ(r)〉 = ψ(r) may be used. This theoretical method was developed in

the early 1960s independently by Eugene Gross and Lev Pitaevskii for weakly interacting BECs

[82, 83, 84]. The mean-field macroscopic wavefunction obeys a Schrodinger-like equation with a

third order nonlinear term, usually called Gross-Pitaevskii equation (GPE), which will be derived

in Chapter 2.3.

The main advantages of BECs include a very good isolation from environmental influence,

high tunability and very precise controllability of almost all parameters [85]. BECs are particularly

good at simulating other physical systems, or quantum simulators as suggested by Feynman [86],

and testbeds for diverse theories. For example, analogue black holes can be created in BECs [87]

12

and the corresponding Hawking radiation with phonon replacing photon can be observed [88, 89].

BECs in optical lattices can also simulate various condensed matter systems such as quantum

phase transitions [90] and Anderson localizations [91]. Moreover, the nonlinearity in BECs allows

the study of soliton dynamics [92, 93, 94]. An experimental test of BECs under microgravity

in a drop tower [95] suggest the possibility of probing the boundary between general relativity

and quantum mechanics. The potential of entangling large numbers of particles in BECs can find

applications in quantum metrology such as using spin squeezing to surpass standard quantum limit

[96, 97]. Mathematical models such as effective negative mass can also be engineered with spin-

orbit coupling [98].

A reason for the high controllability is because of many available choices of BEC systems.

Today, BEC has been realized in many different systems, including almost all alkali atoms, lithium

(7Li), sodium (23Na), potassium (39K, 41K), rubidium (85Rb, 87Rb), caesium (133Cs) [10]. Other

atomic BECs include hydrogen (1H), chromium (52Cr), ytterbium (170Yb, 174Yb), the metastable

excited state of helium (4He∗), Calcium (40Ca), Strontium (84Sr, 86Sr, 88Sr), Dysprosium (164Dy),

and Erbium (168Er). It is also possible to have molecular BECs, for example, bosons formed by

a pair of fermionic atoms, such as 6Li and 40K molecules. Since photons are also bosons, the

BEC transition for photons can also be observed [99]. Room-temperature BECs can be created for

exciton-polaritons because of their very low effective mass and resulting high transition tempera-

ture [100].

The typical scales of time, length, temperature, and energy of dilute atomic BECs can span

several orders of magnitude. For atomic BECs, the lifetime τ can be as long as τ ∼ 10ms−100s,

where τ ∼ 100ms for most experiments. A typical experiment usually involves between 103 to

106 atoms, and the transition temperature Tc is between 100nK to 1000nK. Near Tc, only a small

fraction of particles are in the condensate, while the other particles are in the thermal component.

To have a pure condensate, a much lower temperature T must be used. For instance, at T ∼ 0.1Tc,

the thermal component will be about∼ 0.1% and can be ignored. It is worth emphasizing that no

13

cryogenic equipment is typically used in the experiments, so such cold condensates exist with a

room temperature background. With alternative technique, the temperature of a condensate can be

in the low picoKelvin regime [101]. The size of the condensates can be of order `∼ 0.1µm−10µm

corresponding roughly to a trapping angular frequency ω ∼ 10Hz− 10kHz. The length can be

adjusted independently in all directions, such that a strong trapping along one direction can reduce

the system to an effective 2D BEC system of disk shape, and similarly for a 1D system with

cigarette shape. This translates to a peak density ρ ∼ 1019m−3− 1022m−3, which is much lower

than the density of air ∼ 1025m−3. Higher densities are hard to achieve because the loss from

three-particle collisions grows as ρ3, so the lifetime becomes short.

Atoms are particles with many internal states that can be controlled by light frequencies from

optical to microwave, as well as electric and magnetic fields. For dilute atomic gases at such low

density, the two-particle interactions can be well described by a single s-wave scattering length as

that is independent of the details of the collision. The nonlinear interaction parameter U in the GPE

is proportional to as, which can be controlled by Feshbach resonances in real time [69, 102]. By

tuning the magnetic field near a Feshbach resonance, all values of as can be achieved theoretically,

including positive as > 0 for repulsive interaction, negative as < 0 for attractive interaction, and

as = 0 for no interaction. It has been demonstrated experimentally that this can be done over many

orders of magnitude [103]. For Rubidium, U/h = 6× 10−17Hz/m3, together with the density

discussed above, ρU/h∼ 103Hz−106Hz. The effect of nonlinearity is significant when the term

is comparable or higher than the scale of the trapping frequency or other scale presented in the

system such as kinetic energy.

Another setup typically considered is BECs with more than one type of boson created by mix-

ing different types of atoms. Alternatively, atoms with different hyperfine states are distinguish-

able bosons, so creating two-component BECs with the same atoms is possible. In this setup, the

atoms with different hyperfine states can be inter-convertible during experiments. Studying two-

component BECs are the important part of my proposals as will be discussed in Chapter 2, Chapter

14

4, and Chapter 5.

1.4 Outline of the thesis

This thesis includes three main projects that I have studied. They are chimera knots in 3D, macro-

scopic spin cat states in BECs, and mediated nonlocal hopping in BECs. The first two projects

provided me with the theoretical foundation for the last one. Since knowledge from several differ-

ent fields is needed to understand the results, the relevant background is provided in Chapter 2. It

includes the dynamics of various oscillators, including quantum, classical, and nonlinear. The two

main differential equations, CGLE and GPE, are introduced. This is followed by the mechanism

of the nonlocal diffusive coupling, the nonlocal hopping, and the basics of BECs.

In Chapter 3, I present the new discovery of stable chimera knot states as published in [8].

Prior to my work, knots were not known to be stable in oscillatory media, nor were such non-

trivial chimera patterns known to exist in 3D. As my simulations show, the stability depends on the

nonlocal coupling. I show the properties, structures, and dynamics for Hopf links and trefoils with

good 3D visualization. The same conclusion holds for simple, complex, and chaotic oscillators. In

complex oscillatory media, we can even observe the synchronization defect sheet for the first time.

In Chapter 4, I present my proposal for creating macroscopic cat states in two-component BECs

using the Kerr nonlinearity as published in [12]. It was unclear how large cat states can be in such

systems. In order to increase the nonlinearity and lifetime, we proposed to use strong trapping of

the smaller BEC component and Feshbach resonance respectively. We analyzed the loss and other

experimental imperfections and concluded that cat states involving hundreds of atoms should be

possible.

In Chapter 5, I present a new mediating mechanism that can result in mediated nonlocal hop-

ping, which is analogous to the effective nonlocal interaction between charged particles mediated

by an electromagnetic field. In my scheme, due to the additional mediating channel without energy

barrier, particles can bypass all energy barriers in the original system by jumping into the mediat-

15

ing channel, and can thus reach much further distances. My derivation shows that this approach

allows independently adjustable on-site nonlinearity, hopping strength and range. This nonlocal

hopping model is interesting also because further results show that it is the first known conservative

Hamiltonian system showing chimera states. To show that it is more than a mathematical model,

I also analyze the possibility of implementing mediated hopping in BECs and conclude that it can

be done using current technology. Moreover, simulations show strong evidence of the presence of

chimera states in the BEC system. The mechanism was discovered while I was attempting to search

for chimera states in quantum systems. After dozens of trials of different dynamical equations, I

found suitable equations and immediately realized that they correspond to a two-component GPE

because of the similarity with diffusive coupling. I also recognized that they represent the Bose-

Hubbard model [13, 104] with nonlocal hopping. A mean-field treatment is used in this thesis and

the full quantum treatment will be the future work.

In Chapter 6, I summarize the main results presented in this thesis and discuss the possible

future directions.

16

Chapter 2

Theoretical background

This chapter is comprised of three main sections with the theoretical background for the next

three chapters. Sec. 2.1 is devoted to the basis of the nonlinear dynamic systems and formulation

in phase spaces. I will review various types of oscillators, including both classical oscillators,

quantum oscillators, and nonlinear oscillators. The mechanism of creating cat states based on Kerr

nonlinearity will also be introduced. Sec. 2.2 introduces the diffusive coupling between oscillators

which gives the complex Ginzburg-Landau equation (CGLE). I will then discuss the mechanism

of the nonlocal diffusive coupling and introduce the nonlocal CGLE. The analogue mechanism of

nonlocal hopping is derived. Sec 2.3 is dedicated to the Bose-Einstein Condensates. Starting from

an interacting BEC, the Gross-Pitaevskii equation (GPE) is derived as the mean-field of a quantum

field equation. Moreover, the nonlinear Kerr Hamiltonian of BECs can be obtained with a single-

mode approximation of the same quantum field equation. Finally, I will give the mathematical

formulation of two-component BECs, which will be used to derive the nonlocal hopping in Chapter

5.

2.1 Oscillators and phase space dynamics

Oscillators are one of the fundamental concepts across all branches of physics and describe many

systems existing in nature with regular bounded motion. This is particularly true for the simple

harmonic oscillator, which happens everywhere as a result of the linear approximation. In this

section, various types of oscillators are introduced, including simple harmonic oscillators, nonlin-

ear oscillators, and self-sustained oscillators. Both classical and quantum oscillators are discussed.

Specifically, the nonlinear Kerr effect in quantum systems provides a method to create Schrodinger

cat states, which will be used in Chapter 4.

17

2.1.1 Simple harmonic oscillators and nonlinear oscillators

Classical Harmonic Oscillator: A classical simple harmonic oscillator (SHO) is given by the

Hamiltonian

H (x, p) =1

2mp2 +

12

mω20 x2, (2.1)

where m is the mass and ω0 is the natural angular frequency, with two canonical variables, canoni-

cal coordinate x and canonical momentum p. The corresponding dynamical equations can be found

by the Hamilton’s equation

x =∂H

∂ p=

pm, (2.2)

p = −∂H

∂q=−mω

20 x, (2.3)

which is linear. The solution can be solved directly using the eigenvalue method, which shows a

rotation around a circle with constant angular speed ω0, as shown in Fig. 2.1. Alternatively, those

two equations can be combined to a second order linear differential equation x = −ω20 x, which

gives the same solution.

The same equation can be represented in a different canonical pair of action variable I =

∮pdq/2π and angle variable θ [105]. The transformation for the SHO can be written as

I =1

2mω0p2 +

12

mω0x2, (2.4)

θ = tan−1(mω0x/p), (2.5)

and the Hamiltonian becomes

H (I,θ) = ω0I. (2.6)

Similarly, the dynamical equations can be obtained by using Hamilton’s equation, which gives

I =−∂H /∂θ = 0 and θ = ∂H /∂ I = ω0. Hence, I(t) is a conserved quantity, which is propor-

tional to the energy in this system, and the phase θ(t) is increasing over time. The quantity I is

proportional to the number of particles in quantum mechanics, as will be clear soon.

18

(a) Simple harmonic oscillator (b) Simple nonlinear oscillator (c) Self-sustained oscillator

Figure 2.1: Dynamics in a phase space with the bright spots indicate two different initial con-ditions, and the faint spots indicate the final states. (a) Simple harmonic oscillator. Phase gainis independent of the amplitude. (b) Simple nonlinear oscillator. The phase gain depends on theamplitude. (c) Self-sustained oscillator. Different initial conditions get closer and closer to thelimit circle over time.

An alternative convenient representation is given by the transformation

z =1√2(√

mω0x+ i1√mω0

p), (2.7)

z∗ =1√2(√

mω0x− i1√mω0

p). (2.8)

The Hamiltonian can be rewritten as

H (z, iz∗) = ω0|z|2 = ω0z∗z. (2.9)

With z and iz∗ treated similar to conjugate variables, the dynamic equation can be found by z =

∂H /∂ (iz∗), giving

iz(t) = ω0z, (2.10)

and the solution

z(t) = z(0)e−iω0t . (2.11)

This implies that the phase points with different initial conditions gain the same phase over time

as shown in Fig. 2.1a.

19

Simple nonlinear oscillator: There are various types of nonlinear oscillators. The simplest non-

linear oscillator is given by the third order nonlinearity, or a quartic term in the Hamiltonian

H (z, iz∗) =12

χ|z|4. (2.12)

Again, the dynamical equation can be obtained by using Hamilton’s equation, as

iz(t) = χ|z|2z. (2.13)

To solve this equation, we note that the action I = |z|2 is time independent because

dIdt

=ddt(z∗z) =

dz∗

dtz+ z∗

dzdt

=1−i

(|z|2z∗)z+1i(|z|2z)z∗ = 0, (2.14)

which means the energy H = χI2/2 is also conserved. Hence, using |z(t)|2 = |z(0)|2, the equation

can be rewritten as

iz(t) = χ|z(0)|2z(t). (2.15)

The solution is

z(t) = z(0)e−iχ|z(0)|2t , (2.16)

which behaves the same as the SHO, except now the angular speed ω depends on the amplitude as

ω = χ|z(0)|2. Therefore, initial points with different amplitudes will lead to different phases at a

later time as shown in Fig. 2.1b.

Typically, a linear term also exists in the system even when the expansion is done to the lowest

order nonlinear as

iz(t) = ω0z+χ|z|2z. (2.17)

It can be shown easily that this equation reduces to Eq. (2.13) using the co-rotating frame as

z→ ze−iω0t . In such a frame, the constant rotation generated by the linear term can be eliminated.

The equation above is the simplest nonlinear oscillator because it has the lowest order nonlinear

term for the system with the phase symmetry (or global U(1)-gauge symmetry) z→ zeiθ for all

θ . This is a symmetry that often exists in fundamental physics and conservative systems. Without

20

this symmetry, the second order nonlinearity can exist. To make it clear, if we assume a general

Hamiltonian with the form

H(z, iz∗) = ∑w1=z,z∗

Kw1w1 + ∑w1,w2=z,z∗

Kw1w2w1w2 + ∑w1,w2,w3=z,z∗

Kw1,w2,w3w1w2w3

+ ∑w1,w2,w3,w4=z,z∗

Kw1,w2,w3,w4w1w2w3w4 + ......, (2.18)

substitute the symmetry above and compare with the original one, all odd power terms have to be

zero. For each even power term, only one combination remains. Hence, the general Hamiltonian

that preserves the phase symmetry is:

H = α|z|2 + β

2|z|4 + γ

3|z|6..., (2.19)

More generally, the system that preserves the gauge symmetry can be written as

H = f (|z|2). (2.20)

Self-sustained oscillator: Instead of a pure nonlinear oscillator that conserves energy as discussed

above, a simple non-conservative nonlinear system can be described by the normal form of a Stuart-

Landau oscillator [2]

z(t) = z− (1+ ib)|z|2z. (2.21)

It is a simple self-sustain oscillator representing a broad class of system with z(t) = (ar + iai)z−

(br + ibi)|z|2z and ar,br > 0. This equation can be reduced to the Eq. (2.21) by going into a co-

rotating frame, rescaling the time t and rescaling the amplitude z. The dynamics are illustrated in

Fig. 2.1c with all phase points trending towards the unit circle |z|= 1. It can be understood easily

by rewriting the variable z(t) = A(t)eiθ(t) in terms of amplitude A(t) and phase θ(t). Substituting

back, the resulting coupled equations are

A = A−A3, (2.22)

θ = −bA2, (2.23)

21

If the amplitude is too small A < 1, then the amplitude will increase over time as A > 0 and vice

versa. So, the equilibrium condition A = 0 gives A = 1. Any small perturbation will eventually

disappear and the system is stable with small noise. Also, the phase advances depending on the

amplitude. In the equilibrium system, it oscillates with a constant angular frequency ω = θ =−b.

Since the dynamics is always attracted to the unit circle |z|= 1 and is oscillating, hence, the name

self-sustained oscillator. The closed loop attractor, |z| = 1 here, is referred as the limit circle. For

the change of the quantity I = |z|2, we have

dIdt

=dz∗

dtz+ z∗

dzdt

= 2|z|2−2|z|4 = 2I−2I2, (2.24)

which is consistent with the amplitude equation above. The quantity I can be interpreted as the

number of particles, so the first term corresponds to the pumping of particles into the system, and

the second term corresponds to the nonlinear particle loss.

Complex oscillators: In general, oscillator dynamics exist in any systems with a well-defined

phase θ . Consider a system with two bounded dynamic variables X and Y , then the phase may be

defined as

θ(t) = tan−1(

Y (t)−Y0

X(t)−X0

), (2.25)

if X(t)− X0 and Y (t) = Y0 are not equal to zero simultaneously. (X0,Y0) is a reference point,

which is a center in case of limit cycle dynamics. For the systems with more than two dynamic

variables, there may have a two-dimensional submanifold that the dynamics behave in a way that θ

is well defined, in particular, near the Hopf bifurcation point. An example is the specially designed

Rossler model [106] with three variables X(t), Y (t), Z(t) with differential equations

X(t) = −Y −Z,

Y (t) = X +aY,

Z(t) = b+Z(X− c), (2.26)

where a, b, c are the control variables and the phase can be defined in the XY plane with (X0,Y0) =

(0,0). Therefore, in general, the phase can be well-defined for much broader systems such as the

22

Figure 2.2: Formation of a cat state. The plots of Husimi Q-function of a coherent state underKerr nonlinearity are shown. Time increase uniformly from t = 0 to t = τc, from left to right andfrom top to bottom.

chemical, biological and neural systems [25]. If two or more such subsystems are put together with

some form of coupling, then the relative phase dynamics can be interesting and allows the study of

the synchronization between subsystems.

2.1.2 Quantum oscillators, Kerr nonlinearity and cat states

Quantum harmonic oscillator: The quantum harmonic oscillator behaves similarly to the classical

counterpart in the limit with a large number of particles. The formulation can be done using the

first quantization rules x→ x, p→ p = −ih∂x, or replacing the Poisson bracket x, p = 1 by

23

commutator [x, p] = ih, or replacing z→√

ha in the formulation in previous section. Or simply

start with the commutation relation of annihilation operators a with commutator [a, a†] = 1.

With this definition, we can write down the Hamiltonian for the quantum harmonic oscillator:

H =

(n+

12

)hω, (2.27)

with the Planck constant h, the trapping frequency ω and the number operator n = a†a. The

dynamics is given by the Schrodinger equation

ih∂

∂ t|ψ〉= H |ψ〉, (2.28)

which can be easily solved using the eigenvalue equation n|n〉= n|n〉 with Fock basis |n〉, and the

solution is

|ψ(t)〉= ∑n

cne−iEnt/h|n〉, (2.29)

with the eigenvalue En = (n+ 1/2)hω . Note that the global phase is unimportant, so it can be

written simply as

|ψ(t)〉= ∑n

cne−inωt |n〉, (2.30)

A special state called coherent state |α〉 is known as the most classical state because the dynam-

ics resemble the oscillatory behavior of the classical counterpart [47]. It is defined as the eigenstate

of the annihilation operator as

a|α〉= α|α〉, (2.31)

with associated eigenvalue α = |α|eiθ , which is a continuous complex number with amplitude |α|

and phase θ . Hence, it gives a quantum-classical correspondence, say for the electromagnetic field,

where the state |α〉 has the corresponding amplitude α , phase θ and intensity |α|2. The coherent

states are also states with the minimum uncertainty. It can be written in terms of the Fock basis as

|α〉= e−|α|2/2

∑n

αn√

n!|n〉, (2.32)

24

which is normalized 〈α|α〉 = 1. The probability of detecting n photons is therefore given by the

Poisson distribution

P(n) = |〈n|α〉|2 = e−|α|2 |α|2n

n!. (2.33)

The dynamics of the coherent states can be found using the Heisenberg picture:

ih∂t a =−[H , a] = hω a, (2.34)

with solution

a(t) = a(0)e−iωt , (2.35)

and similarly for the eigenvalue

α(t) = α(0)e−iωt , (2.36)

So, the motion of the coherent state is periodic with angular frequency ω .

To compare the classical and quantum harmonic oscillator, a phase space representation of the

states is required. One such common representation used in quantum optics is given by the Husimi

Q-function:

Q(β ) =1π〈β |ρ|β 〉, (2.37)

with a complex number β and the corresponding coherent state |β 〉. The density operator ρ =

∑i pi|ψi〉〈ψi| with probability pi describes the statistical mixture of quantum states, and takes the

form ρ = |ψ〉〈ψ| for a pure state |ψ〉. A density operator description is required if there is uncer-

tainty about the quantum states such as decoherence or experimental uncertainty. The Q-function

is essentially a measure of how close the state ρ is to the coherent state with β . It is normal-

ized as∫

Q(β )dβ = 1, so Q(β ) may be interpreted as the probability to measure the state with

value β . Note that β is a complex number and is not directly measurable, which corresponding to

the fact that |β 〉 forms an over-complete basis. Experimentally, Q(β ) can be reconstructed using

tomographic techniques. In particular, the Q-function of the coherent state ρ = |α〉〈α| is

Q(β ) =1π|〈α|β 〉|2 = 1

πe−|α−β |2, (2.38)

25

which is Gaussian distributed as shown in the first plot in Fig. 2.2. The existence of the width of

order 1 independent of |α| is due to the minimum uncertainty principle. Also, under the harmonic

oscillator Hamiltonian H above, the dynamics of the Q-function is

Q(β , t) =1π|〈α(t)|β 〉|2 = 1

πe−|α(t)−β |2 =

1π

e−|αe−iωt−β |2. (2.39)

It means that the Gaussian bump rotates along the circle with radius |α| with angular frequency ω

in the phase space. If the amplitude |α| becomes large enough, the distribution may be treated as a

point. Hence, the system behaves similarly to the classical harmonic oscillator.

Nonlinear Kerr effect: Similar to the classical system, the lowest order nonlinearity of the system

preserving the gauge symmetry is given by

H = hχ n2. (2.40)

where n is the number operator, and χ is the Kerr nonlinearity. This Hamiltonian is usually known

as the Kerr Hamiltonian in quantum optics, which is originated from the refractive index change of

the material when a light with strong intensity is passing through. Similarly, in BEC, similar term

exist because of the two-particle interaction. A state evolves under this Hamiltonian as

|ψ(t)〉= ∑n

cne−iEnt/h|n〉= ∑n

cne−in2χt |n〉, (2.41)

which has a recurrence time χtr = 2π in which the state returns back to the same initial state

because e−in22π = 1 for integer n.

This Hamiltonian can be used to generate a cat state |CAT 〉 ∼ |α〉+ i|−α〉, if the system starts

with the coherent state |ψ〉 = |α〉 at t = 0, as originally proposed in [59]. At time χt = π/2, the

factor becomes e−in2χt = 1,−i,1,−i, ... for n = 0,1,2,3, .... Multiplying by the global phase eiπ/4,

26

which rotates the phase space by π/4, the series becomes eiπ/4e−in2χt = (1+(−i)n)/√

2, and

|ψ(t)〉 = e−iH t/h|α〉 (2.42)

= e−|α|2/2

∑n

αn√

n!(1+ i(−1)n)√

2|n〉 (2.43)

=1√2

e−|α|2/2

∑n

(αn√

n!+ i

(−α)n√

n!

)|n〉 (2.44)

=1√2(|α〉+ i|−α〉) , (2.45)

The state becomes a cat state at a special time called cat time τc = π/(2χ). The dynamics of a

cat state formation can be visualized in the phase space using the Q-function as shown in Fig. 2.2.

This construction mechanism is purely quantum mechanical and has no classical counter part.

2.2 Field equations and nonlocal coupling

Oscillations often refer to spatial motion, e.g. for a spring-mass system or a pendulum. But

oscillators can also exist in phase space without any reference to spatial coordinates. For example,

the electromagnetic field can oscillate in any spatial point. This leads to the concept of a physical

field, which describes the assignment of quantities at all points in space and time, for example, the

electromagnetic field and the gravitational field. Other systems can often be well approximated

by field equations, when the scale of interest is much larger than the discreteness of the smallest

system scale, such as the continuum approximation in fluid dynamics for the tiny particles. In this

section, I introduce the field equations and the nonlocal coupling that will be used in Chapter 3 and

Chapter 5. In particular, I try to emphasis the similarity between the nonlocal diffusive coupling

and nonlocal hopping, which provides the motivation to study the nonlocal hopping in Chapter 5.

2.2.1 Local coupling and complex Ginzburg-Landau equation

A system with oscillators coupled locally with their nearest neighbor can be written as reaction-

diffusion system [16]

∂tX(r, t) = F(X)+D∇2X, (2.46)

27

for a dynamic variable X. The vector field F represents a local limit-cycle oscillator of X. The

coupling of local oscillators is through the diffusion ∇2X with diffusion matrix D. Note that D

usually contains no imaginary part, but for, say, quantum systems, a similar term like this can have

complex numbers. Suppose the self-sustained Stuart-Landau oscillators are used

F(z) = z− (1+ ib)|z|2z, (2.47)

with z a complex number, then the field equation can become

z(t) = z− (1+ ib)|z|2z+(1+ ic)∇2z, (2.48)

with a choice on the diffusion. This is called the complex Ginzburg-Landau equation (CGLE)

which describes a broad dissipative system near the Hopf bifurcation. The full derivation of CGLE

for any oscillators near the Hopf bifurcation can be found in, say, [16]. Similar can be done in the

Rossler model with the local oscillators given by Eq. (2.26). On the other hand, if non-dissipative

isolated oscillators are used

F(z) =−iu|z|2z, (2.49)

then the field equation is

iz(t) = u|z|2z−κ∇2z, (2.50)

with a pure imaginary diffusion. It is often called the nonlinear Schrodinger equation. Note that this

equation can be treated as a special case of the CLGE by setting a→−∞ and b→ ∞. In addition,

it is also a special case of the Gross-Pitaevskii equation (GPE) with no trapping potential (see

next section). In literature, GPE is sometimes mixed with nonlinear Schrodinger equation. In this

thesis, I will refer GPE exclusively to the mean-field equations of BECs. Note that the properties

of these equations are significantly different. CGLE describes an open system that the energy and

particles can be exchanged, while nonlinear Schrodinger equation and GPE are conservative.

28

2.2.2 Nonlocal diffusive coupling

Similar to the direct local coupling discussed above, localized oscillators can be coupled with each

other through a mediating channel. It can be illustrated by the following equation [6]

∂tX(r, t) = F(X)+ kg(S), (2.51)

τ∂tS(r, t) = D∇2S−S+h(X), (2.52)

where F(X) describes a self-sustained oscillator with the dynamic variable X(r, t). k is the control

parameter of the feedback strength from the the channel with the form g(S). S(r, t) is a real variable

that decays over time plus a source term h(X). The spatial coupling dynamics is the diffusion

in the mediating channel with diffusion constant D > 0. τ in the second equation sets the time

scale. With a large separated time scales, adiabatic elimination can be used to derive the nonlocal

diffusive coupling as done in [6].

Here, I will derive a similar alternative form which can highlight the similarity between the

nonlocal diffusive coupling and the mediated nonlocal hopping derived in the next subsection. Let

the local dynamic be the Stuart-Landau oscillator F(z), and the equations

∂tz(r, t) = z− (1+ ib)|z|2z+ kξ , (2.53)

τ∂tξ (r, t) = D∇2ξ −ξ +h0z, (2.54)

where both z and ξ are complex dynamical variables. The second equation is a mediating channel

that propagates the perturbation between oscillators and can be considered as two independent

diffusion equations for the real and imaginary part of ξ . Note that the second equation is linear,

so it has an exact solution. Here we consider only the adiabatic limit τ = 0. In this limit, the

mediating channel becomes

0 = D∇2ξ −ξ +h0z. (2.55)

Suppose the Fourier transform of the dynamical variables are

z(q, t) = F [z(r, t)] = (1/(2π)d)∫

dre−iq·rz(r, t) (2.56)

29

and similar for ξ (q, t) = F [ξ (r, t)]. So, the differential equation becomes an algebraic equation

0 =−q2Dξ − ξ +h0z, (2.57)

and the solution for ξ is

ξ = h01

1+Dq2 z. (2.58)

The inverse Fourier transform gives the convolution

ξ = h0G(r)∗ z(r, t) = h0

∫dr′G(r− r′)z(r′, t), (2.59)

assuming isotropic and translational invariant system with unbounded domain, where

G(r) = F−1[

11+Dq2

]. (2.60)

The explicit solutions in different dimensions are

G1D(r) =1

2Re−r/R, (2.61)

G2D(r) =1

2πR2 K0

( rR

), (2.62)

G3D(r) =1

8πR31r

e−r/R, (2.63)

where r = |r| and R =√

D is the effective radius and r = |r|. K0 is the modified Bessel function of

the second kind. Substituting the solution back to Eq. (2.53), we have

∂tz(r, t) = z− (1+ ib)|z|2z+ kh0

∫dr′G(r− r′)z(r′, t), (2.64)

and can be rewritten as

∂tz(r, t) = z− (1+ ib)|z|2z+K(1+ ia)∫

dr′G(r− r′)(z(r′, t)− z(r, t)

). (2.65)

This equation is called the nonlocal CGLE and can have stable chimera states. Specifically, the

first observation of chimera core patterns appears in this model [7].

30

2.2.3 Nonlocal hopping with mediating channel

Considering a similar set of differential equations where simple nonlinear oscillators are used

instead of self-sustained oscillators:

iψ1(r, t) = U |ψ1|2ψ1 +Ωψ2, (2.66)

iψ2(r, t) = −κ∇2ψ2 +∆2ψ2 +Ωψ1. (2.67)

where ψ1 and ψ2 are dynamic variables representing wavefunctions. U is the nonlinear coefficient,

Ω is the Rabi frequency, ∆2 is the detuning, and κ is the inverse mass. All control variables are real

numbers. The equations are related to mean-field GPE of two-component BEC which is derived

in next section. Note that this model without kinetic energy term is just a mathematical model,

and a realistic system will be considered in Chapter 5. It, however, greatly simplifies the detail

and allows highlighting the mechanism for the effective nonlocality. The second equation is the

mediating channel that propagates the perturbation and is similar to the equation in the previous

section with an extra i in front of the time derivative. If adiabatic elimination is used τ = 1/∆2 = 0,

then the mediating channel becomes

0 =−κ∇2ψ2 +∆2ψ2 +Ωψ1. (2.68)

Let R =√

κ/∆2 and use Fourier transform, so

0 = q2R2ψ2 + ψ2 +

Ω

∆2ψ1, (2.69)

This equation is exactly the same as before, so it has the same solution as

ψ2(r, t) =−Ω

∆2G(r)∗ψ1(r, t), (2.70)

with G(r) is the same in previous section. Substituting back to the non-mediating channel, we have

iψ1(r, t) =U |ψ1|2ψ1−Ω2

∆2

∫drG(r− r′)ψ1(r′, t) (2.71)

Furthermore, this system has another set of solutions. Since the physical meaning of ∆2 is

the detuning, which can be negative. The mediating equation with adiabatical elimination can be

31

rewritten as

0 =−∇2ψ2−

1R2

0ψ2 +

Ω

κψ1, (2.72)

where R0 =√−κ/∆2 and the corresponding solutions for the Sommerfeld radiation boundary

condition are

G1D−(r) =iR0

2e−ir/R0, (2.73)

G2D−(r) =i4

H0(r/R0), (2.74)

G3D−(r) =1

4πre−ir/R0, (2.75)

where H(1)0 is the Hankel function. These equations represent wave-like solutions for negative

∆2 < 0, while compared with the confined solutions for positive ∆2 > 0.

2.3 Bose-Einstein Condensates

Bose-Einstein Condensates are states of matter where all particles condense to the same single-

particle state. This happens by cooling non-interacting systems below the BEC transition temper-

ature. As long as the interaction is not too strong, BECs can still exist. Hence, the property that

all particles are in the same single-particle state allows simple mathematical descriptions of the

systems, in comparison with strongly interacting condensed matter systems. Depending on setups,

BECs can be used to study a large variety of interesting physics. In this section, I introduce the

weakly interacting BECs at zero temperature. The mathematical descriptions of two setups, the

Kerr effects, and the two-component BECs, used in Chapter 4 and Chapter 5 are discussed.

2.3.1 Gross-Pitaevskii equation

We consider BECs at zero temperature so that all thermal excitations can be ignored. In the ideal

situation without interaction, all bosons are in the same single-particle state. When the interaction

is turned on, some particles may be kicked out from the condensates. The Hartree mean-field

approach [10] can be used in the weak interaction regime by assuming that the full wavefunction

32

is the symmetrized product of a single-particle wavefunction φ(r). So the full wavefunction of

N-particle system is

Ψ(r1,r2, ...,rN) =N

∏i=1

φ(ri), (2.76)

with normalization∫

dr|φ(r)|2 = 1. (2.77)

For N identical non-interacting bosons, the Hamiltonian is

H0 =N

∑i

H0,i =N

∑i

[p2

i2m

+V (ri)

], (2.78)

where H0,i is the Hamiltonian of an individual particle, pi is the momentum operator, m is the mass

of the particle, and V (ri) is the external potential. The independence and additivity of Hamiltonian

allow the use of separation of variables, so φ0,i(ri) can be obtained by solving

H0,iφ0,i = εiφ0,i, (2.79)

with the eigenenergy εi. Hence, the total wavefunction can be obtained by combining φ0,i(ri).

Since all single-particle wavefunction are the same, so φ0,i(ri) = φ0(ri).

Now, suppose a dilute atomic gas of bosons is considered. In this system, the dominant interac-

tion are the two-particle collisions of the form U0δ (r− r′) where δ (r) is the Dirac delta function.

The interaction strength is U0 = 4π h2as/m with the s-wave scattering length as. Therefore, the

system is given by the Hamiltonian

H = H0 +U0 ∑i< j

δ (ri− r j), (2.80)

where the summation is taken over all pair of particles. The mean-field energy E of the state Ψ can

be calculated as

E = 〈Ψ|H |Ψ〉= N∫

dr[

h2

2m|∇φ |2 +V |φ |2 + N−1

2U0|φ |4

]. (2.81)

The term N(N− 1)/2 counts the number of pair of particles and can be approximated by N(N−

1)/2≈ N2/2, with the term of order 1/N being ignored.

33

The wavefunction of the condensate, which is sometimes referred to as the order parameter,

can be defined as ψ(r) =√

Nφ(r). Hence, the total number of particle is N =∫

dr|ψ|2 and the

particle density is n(r) = |ψ(r)|2. So the energy can be rewritten as

E[ψ] =∫

dr[

h2

2m|∇ψ|2 +V |ψ|2 + 1

2U0|ψ|4

]. (2.82)

By minimizing the E− µN with respect to ψ∗, where µ is a Lagrange multiplier, we can obtain

the time-independent Gross-Pitaevskii equation (GPE)

µψ(r) =− h2

2m∇

2ψ +V ψ +U0|ψ|2ψ. (2.83)

This is basically the Schrodinger equation with an extra nonlinear term that originated from the

two-particle collision. The eigenvalue µ has the meaning of chemical potential.

The corresponding time-dependent GPE is

ih∂tψ(r, t) =− h2

2m∇

2ψ +V ψ +U0|ψ|2ψ, (2.84)

which describes the dynamics of the BEC in the mean-field limit. This equation can also be

obtained by the variational derivative ψ = δE/δ (ihψ∗). Note that some particles are in states that

are different from the others due to interactions. So the dynamic equation above is true for the

majority of the atoms except for a small number of particles of order (na3s )

1/2. The fact that not

all particles are in the same state is referred as quantum depletion. For most experiments, it is of

order one percent or less and can usually be ignored.

2.3.2 Kerr nonlinearity in BEC

Using Eq. (2.80), the second quantized Hamiltonian is

H =∫

dr[− h2

2mψ

†∇

2ψ +V ψ

†ψ +

12

U0ψ†ψ

†ψψ

], (2.85)

where ψ(r) is the boson annihilation operator and ψ†(r) is the corresponding creation operator

with the usual commutation relation [ψ(r), ψ†(r′)] = δ (r− r′). The time evolution of the field

34

ψ(r) operator can be obtained by using the Heisenberg equation of motion ih∂tψ = −[H , ψ],

which gives

ih∂tψ(r, t) =− h2

2m∇

2ψ +V ψ +U0ψ

†ψψ. (2.86)

The field operator can be written as ψ(r) =ψ(r)+δψ(r), where ψ(r) is the mean-field and δψ(r)

is the quantum fluctuation. In the mean-field limit ψ(r)→ 〈ψ(r)〉 = ψ(r), where the quantum

fluctuation is ignored, we can obtain the same GPE in Eq. (4.6).

If the interaction is weak and the quantum fluctuation can be ignored, then a BEC can be

described by a single spatial wavefunction φ(r). In this case, the field operator can be rewritten as

ψ(r) = aφ(r), (2.87)

where a is the annihilation operator of the mode φ(r). Substituting this equation back to the

Hamiltonian in Eq. (2.85) can be reduced to

H = ε0n+12

Un(n−1), (2.88)

with

ε0 =∫

dr[

h2

2m|∇φ |2 +V |φ |2

], (2.89)

U = U0

∫dr|φ |4, (2.90)

where n = a†a is the number operator. The linear term can be dropped in a rotating frame, so the

Hamiltonian can be rewritten as

H =12

Un2, (2.91)

which is exactly the Kerr nonlinearity discussed before. It is the analogue of the optical Kerr effects

in ultracold atoms. Hence, it provides a mechanism to create cat states in BECs.

2.3.3 Kerr nonlinearity in two-component BEC

The detection of the cat states requires a reference BEC to read out the phase information. So we

may consider using a two-component BEC. To begin with, we consider a general two-component

35

model given by the Hamiltonian

H = ∑i=1,2

(Hi +

12U ii

)+ U 12 + R, (2.92)

with

H i =∫

dr(

h2

2mi∇ψ

†i (r)∇ψi(r)+Vi(r)ψ†

i (r)ψi(r)), (2.93)

U i j = gi j

∫drψ

†i (r)ψ

†j (r)ψi(r)ψ j(r), (2.94)

R = ∑i=1,2

h∆i

∫drψ

†i (r)ψi(r)+ hΩ

∫dr(

ψ†1 (r)ψ2(r)+ ψ

†2 (r)ψ1(r)

), (2.95)

where mi is the mass, Vi(r) is the potential function, and gi j is the two-particle collision energy

density. The Rabi oscillation term R represents the inter-conversion between the two components

with Rabi frequency Ω and detuning ∆i. The detuning is the frequency difference ∆i = ωi−ω

between the driving frequency and the frequency of the target internal energy level.

We consider the setup with R = 0 here, which means that there is no conversion between the

two components and the number of particles in each component is conserved. Again, assuming

both the BEC components can be described by single-particle modes φi(r), so the field operator

becomes

ψi(r) = aiφi(r). (2.96)

Substituting back into the Hamiltonian above, we have [107]

H = ε1n1 + ε2n2 +12

U11n21 +

12

U22n222 +U12n1n2, (2.97)

with

εi =∫

dr[

h2

2m|∇φi|2 +V |φi|2

], (2.98)

Ui j = gi j

∫dr|φi|2|φ j|2. (2.99)

Since the total number of particles N = n1 + n2 is conserved in the atomic system, so we can set it

to a constant. The above equation can be rewritten in term of n1, with the constant and linear terms

36

dropped, as

H =12(U11 +U22−2U12) n2

1. (2.100)

So in this case, there is also Kerr nonlinearity that can generate a cat state. Note that φ1 and φ2

are usually different even in the ground state for few reasons. For example, the spatial mode of

condensates with intra-atomic interactions can depend on the number of atoms. Also, the inter-

atomic interaction changes the spatial modes of both of them. If all three scattering lengths ai j are

very close, such as for Rubidium atoms, and N1 ≈ N2, then φ1 ≈ φ2 [96] is a good assumption.

2.3.4 Mean-field equation of two-component BEC

The time evolution of the field operators in a two-component BEC can be found by Heisenberg

equation ih∂tψi = −[H , ψi] with H in Eq. (2.92). If the mean field of the operators is taken,

ψi(r)→ 〈ψi(r)〉= ψi(r), then the dynamic equations are

ih∂tψ1(r, t) =

(− h

2m∇

2 +V1 +g11|ψ1|2 +g12|ψ2|2 + h∆1

)ψ1 + hΩψ2, (2.101)

ih∂tψ2(r, t) =

(− h

2m∇

2 +V2 +g12|ψ1|2 +g22|ψ2|2 + h∆2

)ψ2 + hΩψ1, (2.102)

which gives the coupled two-component GPE. Note that the physics is invariant with a constant

energy shift, so one of the detunings ∆1 or ∆2 can be eliminated, say, ∆1 = 0. The trapping potential

V1 and V2 are usually the same, but can be different by using spin dependent potential. This can

significantly change the relative density of both components and so the effective nonlinearity. The

nonlinearity gi j = 4π h2ai j/m depends on the scattering ai j between components i and j. Using

Feshbach resonances, one of the scattering lengths ai j can be usually adjusted freely. Depending

on experiments, both large and small ai j can be used. For large ai j, the corresponding nonlinearity

can be enhanced, while for small ai j, the corresponding three-body loss can be suppressed.

Now, suppose we consider the system with no trapping on the second component, so V2 = 0.

Also, the nonlinear interaction involving ψ2 is negligible, i.e. g12 = g22 = 0. Then the resulting

37

equations are

ih∂tψ1(r, t) =

(− h

2m∇

2 +V1 +g11|ψ1|2)

ψ1 + hΩψ2, (2.103)

ih∂tψ2(r, t) =

(− h

2m∇

2 + h∆2

)ψ2 + hΩψ1, (2.104)

which are the equations used in Chapter 5. Note that the equations describe a real BEC setup in

experiment and should be implementable.

38

Chapter 3

Linked and knotted chimera filaments in oscillatory systems

3.1 Preface

While the existence of stable knotted and linked vortex lines has been established in many exper-

imental and theoretical systems, their existence in oscillatory systems and systems with nonlocal

coupling has remained elusive. Here, we present strong numerical evidence that stable knots and

links such as trefoils and Hopf links do exist in simple, complex, and chaotic oscillatory systems

if the coupling between the oscillators is neither too short ranged nor too long ranged. In this case,

effective repulsive forces between vortex lines in knotted and linked structures stabilize curvature-

driven shrinkage observed for single vortex rings. In contrast to real fluids and excitable media,

the vortex lines correspond to scroll wave chimeras [synchronized scroll waves with spatially ex-

tended (tubelike) unsynchronized filaments], a prime example of spontaneous synchrony breaking

in systems of identical oscillators. In the case of complex oscillatory systems, this leads to a novel

topological superstructure combining knotted filaments and synchronization defect sheets.

The results in this chapter were part of my research and were published in [8]. This work was

started by the observations of knots in 3D simulations of nonlocal Kuramoto model during the

early stage of my PhD. To further study the phenomenon, I wrote the simulation and visualization

programs specifically for this project. The manuscript was written with the guidance and criticism

from Prof. Davidsen.

3.2 Introduction

In natural science, knots and linked structures have attracted attention in various branches as they

are an essential part of many physical processes. This includes real fluids [108], liquid crys-

39

tals [109, 110], Bose-Einstein condensates [111, 112], electromagnetic fields and light [113, 114],

superconductors [115], proteins [116] as well as excitable media [117, 118] and bistable me-

dia [119]. Stable knots and their topological invariants are of particular interest for both theory

and experiments as they play an important role in characterizing and controlling different systems

and their dynamics [120]. This is especially true in excitable media, where linked and knotted fil-

aments of phase singularities can be essential to understand the nature of scroll wave propagation

processes [117, 118, 3, 121], including nonlinear wave activity associated with ventricular fibrilla-

tion and sudden cardiac death [122, 123]. While the wave propagation processes in excitable and

nonlinear oscillatory systems are very similar [16], the existence of such stable knotted and linked

filaments in oscillatory systems has remained elusive. For example, to the best of our knowledge

no corresponding parameter regime has been identified in the complex Ginsburg-Landau equation

(CGLE), which is the normal form of oscillatory media close to the Hopf bifurcation [1, 36]. This

is deeply unsatisfying as the collective behavior, spontaneous synchronization and wave propaga-

tion in oscillatory media and coupled systems of nonlinear oscillators are topics of general interest

with applications across disciplines [16, 25], including the quantum regime [124, 125].

In this paper, we show for the first time that (i) stable knotted and linked filaments do exist

in oscillatory systems, (ii) they do exist under non-local coupling in the underlying dynamical

equations, and (iii) together with synchronization defect sheets they can form novel topological

superstructures. From the Kuramoto model of simple phase oscillators and the CGLE to com-

plex and chaotic oscillatory systems, we find in particular Hopf links and trefoils that persist over

hundreds of thousands of scroll wave rotations for a wide range of parameters. Due to the non-

local coupling, the filaments that make up the long-lived knotted structures are no longer simple

phase singularities as is typical for scroll waves, but instead the filaments correspond to spatially

extended regions in which the oscillators are unsynchronized. This is despite the fact that all oscil-

lators are identical and uniformly coupled. The coexistence of these unsynchronized local regions

with synchronized regions — exhibiting traveling waves in our specific case — is the hallmark of

40

a chimera state [4, 18, 20, 126, 5, 34]. While single ringlike chimera filaments shrink, knotted and

linked filaments generate an effective repulsion that prevents shrinkage and stabilizes the pattern

even in the presence of strong noise. We find that for coupling that is too short ranged (includ-

ing local coupling) or coupling lag that is too small, the repulsion is too weak such that knotted

structures collapse. This is despite the fact that phase twists along the filaments are present, which

have been hypothesized to have a stabilizing effect by themselves [118]. If the coupling between

oscillators is too long ranged and the coupling lag is too large, straight chimera filaments become

unstable in a way reminiscent of negative line tension [123, 127, 128, 129, 130, 131, 132]. This

leads to the breakup of the knotted structures as well.

3.3 Phase oscillators

As the simplest paradigmatic model of an oscillatory system, we first focus on the Kuramoto

model [5, 2, 26]. In this model, θ(r, t) ∈ [−π,π) denotes the state of an oscillator at a spatial point

r and time t. The evolution is governed by

θ(r, t) = ω0 +Kω(r, t). (3.1)

Here, ω0 is the natural frequency of the oscillators and K is the coupling strength. Note that we

can set ω0 = 0 and rescale time Kt → t without loss of generality. Thus, ω is the instantaneous

angular frequency obeying

ω(r, t) =∫

VG0(r− r′)sin[θ(r′, t)−θ(r, t)−α]dr′ (3.2)

where G0(r) is a coupling kernel, α is the coupling lag or phase shift, and the integration is taken

over the whole volume V . The kernel used is a top-hat kernel with coupling radius R

G0(r)∼

1, r ≤ R

0, r > R(3.3)

41

which is normalized as∫

V G0(r)dr = 1. In simulations, the spatial locations are discretized into

r = (xi,yi,zi) with 1 ≤ xi,yi,zi ≤ L taking integer values in the system of linear length L such

that V = L3. Hence, the control parameters of the system are R and α , with finite size effects

determined by L. Extensive simulations have been done using the Runge-Kutta scheme 1 with

both random initial conditions (IC) and specific functions, see the Appendix for details. We use

periodic boundary conditions (BC) here, yet our findings are quite independent of the BC.

The Kuramoto model with nonlocal coupling is known to exhibit chimera states, in which

both synchronized and unsynchronized regions of oscillators can coexist in the same system even

though all oscillators are identical and uniformly coupled. Most studies have been done on the

one dimensional ring and complex networks [5]. In higher dimensions, two qualitatively different

chimera regimes have been identified for the Kuramoto model given by Eqs. (3.1), (3.2), (3.3). For

near global coupling with R ∼ L and large α . π/2, various coherent and incoherent strip, spot,

plane, cylinder, sphere and cross patterns have been observed in two and three dimensions (2D

and 3D) [34, 133]. The other regime involves shorter range nonlocal coupling L R 1 with

smaller α . In 2D geometries, synchronized spiral waves with unsynchronized chimera cores can

appear. They behave like a normal spiral, yet the dynamics in the core is unsynchronized [133, 19,

134, 135]. Similarly, in 3D, regular scroll waves with chimera filaments (or chimera tubes) at their

center — instead of the linelike filaments of phase singularity — have been observed [34].

3.4 Existence of knots

For L R 1 and large effective system size L/R, we observe different stable linked and knotted

scroll waves in the Kuramoto model as shown in Figs. 3.1 and 3.2. To clearly visualize the

chimera tubes and the knotted and linked structures (referred to as knots in the following), one

has to take into account that both phase θ(r, t) and angular frequency ω(r, t) fluctuate a lot in

1 We have tested different time steps using both explicit Runge-Kutta and Euler’s method. The stable topologicalstructures are preserved but the trajectories deviate after a long time. This is expected since a large time step introducesan effective noise. Similar observations have been mentioned in Ref. [118] for excitable media.

42

Figure 3.1: (color online) Chimera nature of knots (α = 0.8, R = 8, L = 200). (a) Snapshotof a Hopf-link shows the unsynchronized phase θ(r, t) on the isosurface of spatially smoothedangular frequency ω(r, t). (b) Mean angular frequency distribution of 〈ω(r)〉, averaged over ap-proximately 10 periods. The tail to the right corresponds to unsynchronized oscillators. (c) Plot of2D cross-section of θ(r) showing the chimera and spiral wave properties. A vertical cut throughboth rings in panel (a), showing four chimera cores. (d) A cut through the far edge of a ring. (e)Slicing of a ring, showing a ring chimera and two chimera cores of the other ring.

space as shown in Fig. 3.1b- 3.1e. Thus, it is helpful to define a local mean angular frequency

ω(r, t) =∫

V G0(r− r′)ω(r′, t)dr. Fig. 3.1a shows a snapshot of the chimera tubes by plotting the

phases of the unsynchronized oscillators for ω(r, t) ≥ const. The presence of scroll waves with

chimera filaments can also be seen directly in the phase field. Selected 2D cross-sections of the

phase field (Fig. 3.1c) show patterns similar to chimera spirals in 2D [19], while other cross-

sections show features that are specific to 3D such as the chimera ring shown in Fig. 3.1e. Note

that the Hopf link and other knots observed generate spherical wave in the far-field. Moreover,

they are not stationary but keep rotating, drifting, and changing their shape over time as shown in

43

the Appendix.

As chimera filaments are associated with scroll waves, phase twists can be present along the

filament [136]. This is visible in Fig. 3.1a, but can be better visualized by considering the local

mean phase field θ(r, t), defined by

ρ(r, t)eiθ(r,t) =∫

VG0(r− r′)eiθ(r′,t)dr′. (3.4)

This is illustrated in Fig. 3.2, for example.

3.5 Phase diagram

For the Kuramoto model given by Eqs. (3.1), (3.2), and (3.3), our numerical simulations allow

us to obtain a phase diagram as a function of α . This is plotted in Fig. 3.2 together with some

of the asymptotic states. At small α , only relatively simple scroll wave structures with straight

chimera tubes are stable. For αK < α < α0, also knots such as 1 twist Hopf links and 3 twist

trefoils (as shown in Fig. 3.2) are stable over hundreds of thousands of scroll wave rotation periods

T 2. More stable structures including helices, ring-tubes and linked triple rings are shown in the

Appendix. For α > α0, knots as well as simple straight tubes become unstable. The evolution

in the former case is shown in Fig. 3.3b. In the latter case, the dynamics of the chimera filament

indicates that a finite wavelength instability of the filament itself occurs such that the filament

grows rapidly (see the Appendix). In both cases, the rapid growth of filaments is accompanied

by fragmentation through collisions leading eventually to an irregular or turbulent-like behavior as

shown in Fig. 3.2. Furthermore, in the same parameter regime near α0 in 2D, chimera spirals are

stable and no irregular pattern is present [19]. All this suggests that the underlying instability is

truly 3D in nature as the negative line tension instability and similar filament instabilities that have

been observed in excitable and oscillatory media [123]. Fig. 3.2 also shows that at even higher

α ∼ π/2, no filament structures can be recognized.2The stability of Hopf links and trefoils in the Kuramoto model has been tested for extended periods of time of at

least t = 1.2×106 (or period T > 105 where T ≈ 11 at α = 0.8) for L = 100 with R = 4, and t = 1.2×105 for L = 200with both R = 4 and R = 8.

44

Figure 3.2: (color online) Phase diagram of the Kuramoto model in 3D as a function of α withnonlocal coupling L R 1. Various knots exist between αK and α0. Top panel shows sta-ble Hopf-link (left), and trefoil (right) as examples. The plots are similar to those in Fig. 3.1a,but smoothed phases θ(r) are used instead. Shadows on the walls correspond to perpendicularprojections of the structures.

The nature of the instability of knots at αK and α0 are significantly different. Below αK , any

knot transforms through one or multiple reconnections into a single untwisted ring which shrinks

and disappears, leading to homogeneous oscillations. As we have tested, all single chimera rings

with a radius of up to 80 shrink and eventually vanish for 0 ≤ α < α0 with no-flux BC (see

Appendix). This together with the stable knots for αK < α < α0 indicates that there is an effective

repulsion between filaments in knots that is sufficient to prevent curvature-driven shrinkage and

stabilize these structures above αK . Below αK , the repulsion is too weak to prevent reconnections.

This mechanism is similar to what has been observed for knots in bistable media [119] and plays

45

Figure 3.3: (color online) Time evolution of Hopf links outside their stability regimes near αK andα0, respectively. (a) α < αK . After α is decreased slightly from above αK to below, the two ringsof the Hopf link merge resulting in a single ring which eventually vanishes. (b) α > α0. After α

is tuned slightly from below α0 to above, one of the rings grows until it collides and reconnects,resulting in a turbulent-like pattern.

an important role in other situations as well [137]. Simulation results show that different knots

have different stability regimes, especially Hopf links are stable over a broader range of α than

trefoils. Therefore, we denote αK in the following as the point at which Hopf links disappear.

3.6 Dependence on R, L, and geometry

Numerical simulations for 4 ≤ R ≤ 12 and 64 ≤ L ≤ 300 show that the phase diagram presented

in Fig. 3.2 is independent of the specific choice of R and L as long as L R 1. Specifically,

αK ≈ 0.61 and α0 ≈ 0.90 with uncertainty ±0.02. The condition L R ensures that finite size

46

effects do not play a significant role as the size of stable knots and the wavelength scale with

R [19, 134]. For example, we find that stable Hopf links cease to exist for L/R . 16. Also, if the

effective system size is too small, more complex knots tend to decay into simpler ones (see the

Appendix). The condition R 1 is also crucial. We find that for shorter range couplings R < 3

the lifetime of knots is finite 3. Specifically, no stable knots have been observed for local coupling,

R = 1, independent of the IC used to generate Hopf links. This is a consequence of temporal

fluctuations in the s hape of the individual rings within a Hopf link becoming comparable to the

minimum separation between the rings such that the rings merge and disappear (see the Appendix)

— the same behavior as for the instability at αK . We observe qualitatively the same for trefoils.

3.7 Robustness with respect to noise

To further quantify the stability of different topological states, we examine them in noisy environ-

ments. This is modeled by an additional Gaussian phase noise ξ (r, t) in the Kuramoto model

θ(r, t) = ω(r, t)+Dξ (r, t) (3.5)

where 〈ξ (r, t)〉 = 0 and 〈ξ (r, t)ξ (r′, t ′)〉 = δ (r− r′)δ (t− t ′). As shown in Fig. 3.4, Hopf links

and trefoils can survive under noise magnitude as high as D∗ = 0.22. This high robustness under

noise signifies the topological protection of knots. Longer range coupling as quantified by R also

increases the tolerance of local phase noise as shown in Fig. 3.4a.

3.8 Dependence on spatial kernel

In contrast to the top-hat kernel G0, we did not observe stable knots for Gaussian kernels often

considered in the context of chimera states. This together with the existence of a minimal R dis-

cussed above indicates that the range of the spatial kernel is crucial. To substantiate this further,

3For R = 2, the lifetime can vary significantly with the used time stepping of the integrator and becomes longer forshorter ∆t. For example, the lifetime is about t = O(5000) and, thus, less than 1000 scroll wave rotations using 4thorder Runge-Kutta with ∆t = 0.02.

47

0 .1 .2 .3D

0

.5

1

HLT

3 4 5 60

.3

f(a)

R

D∗

.6 .7 .8 .9α

0

.1

.2

.3

StableKnots

Tubes Irre-gular

1/n1

(b)

Figure 3.4: (color online) (a) The tolerable phase noise level for Hopf links (HL) and trefoils(T). f quantifies the fraction of structures persisting after t = 5000 (≈ 450 scroll wave rotations)under noise intensity D. Each point corresponds to an ensemble of 10 realizations. Here, α = 0.8,R = 4, and L = 100. Inset: D∗ denotes the point of f = 0.5 for a Hopf link as a function of R usingα = 0.8, L/R = 16. (b) Phase diagram showing the stable regime of Hopf links for kernel G1 withL R 1. The red dot marks the triple point between all three phases where αK = α0. The errorbars account for different R≥ 4 and L up to L = 300. Note that 1/n1 = 0 corresponds to the top-hatkernel G0 used in Fig. 3.2.

let us consider the kernel G1(r) ∼ e−(r/R)n1 such that G1→ G0 when n1→ ∞ if R < L. Note that

n1 = 2 is a Gaussian, n1 = 1 is an exponential, and n1 = 0 gives global coupling. Simulations show

that if G1 becomes more long-ranged as n1 decreases, the stable regime αK < α < α0 of knots

shrinks as shown in Fig. 3.4b. This is the only effect on the knots as the nature of the associated

instabilities along the boundaries appears unchanged and follows the pattern shown in Fig. 3.3.

The phenomenon is independent of the exact functional form of the kernel (see the Appendix).

3.9 Beyond phase oscillators

The nonlocal CGLE is given by [7]:

A(r, t) = A− (1+ ib)|A|2A+(1+ ia)pA, (3.6)

48

0 1 2 3 4 5 6 7R

0

5

10

15

20N

umbe

rof

latt

ice

site dmin

ξ

(a)

Figure 3.5: (color online) (a) CGLE [(a,b) = (1,0), K = 0.1]: Comparison of minimum separationbetween rings in a Hopf link, dmin, and a measure of spontaneous fluctuations in the ring shape,ξ (see the Appendix for details). Bars indicate the 99% range. (b) Rossler model: Topologicalsuperstructure of a chimera Hopf link with an attached sychronization defect sheet in the perioddoubled regime (see the Appendix for more viewpoints and dynamics).

where the control parameters are (a,b). The nonlocal coupling pA is given by

pA(r, t) = K∫

G0(r− r′)[A(r′, t)−A(r, t)]dr′, (3.7)

and the coupling strength is K. Since the CGLE can be well approximated by the Kuramoto

model in the weak-coupling limit independent of the specific coupling [18, 138], similar results are

expected in certain parameter regimes. Indeed, stable chimera knots with non-constant amplitudes

|A| exist in the vicinity of the parameters (a,b) = (1,0) [7] for K = 0.1 (see the Appendix) and

larger values of K. The lifetimes of knots are longer than 6×105, provided that R 1. All results

discussed above for the Kuramoto model also hold qualitatively for the CGLE. This includes in

particular the break-up of knots for small R. Fig. 3.5a provides a clear rationale why this happens:

The separation between the rings in a Hopf link shrinks with decreasing R such that it eventually

becomes comparable to the amplitude associated with the temporal fluctuations in the shape of

individual rings. This offers an explanation of why no stable knots have been observed in the

CGLE with local coupling. Our findings for the CGLE imply that all oscillatory systems with

49

appropriate nonlocal coupling should exhibit stable knots in some parameter regime near their

Hopf bifurcation.

3.10 Complex oscillatory systems

We also observe stable chimera knots if the uncoupled oscillators are far from the Hopf bifurca-

tion and undergo complex or even chaotic oscillations, requiring at least a three-dimensional local

phase space. A specific example is the Rossler model with nonlocal coupling [139], which exhibits

a phenomenology with many features in common with those observed in complex oscillatory sys-

tems including chemical experiments [140]. It is given by

X(r, t) = −Y −Z + pX ,

Y (r, t) = X +aY + pY , (3.8)

Z(r, t) = b+Z(X− c),

where the control parameters are (a,b,c) and the nonlocal coupling pX(r, t) and pY (r, t) are defined

analogously to Eq. (3.7). For a= b= 0.2, the effective α decreases as c increases [138]. For R 1,

we observe stable chimera knots in the period-doubled regime (c = 3.6) and in the chaotic regime

(c = 4.8) with weak coupling K = 0.05 (see the Appendix). All findings described above for the

other models hold qualitatively here as well. Stable knots only exist if the coupling between the

oscillators is neither too short-ranged nor too long-ranged. For example, we did not observe stable

Hopf links or trefoils for R = 1 or when the kernels were Gaussian in the parameter regimes given

above. Moreover, in the period-doubled regime, synchronization defect sheets (SDS) — the analog

of synchronization defect lines in 2D systems [139, 141] — can be observed for the first time and,

more importantly, connect the different filaments (see Fig. 3.5b, the Appendix). This leads to

another layer of topological structure associated with the knots, making this a unique phenomenon

and adding potentially to their general robustness if multiple knots are present [140].

50

3.11 Discussion and conclusions

Our findings show that knots exist and are stable over a significant range of parameters in various

oscillatory systems with nonlocal coupling as long as the characteristic coupling length of the

kernel is sufficiently large and the tail of the kernel decays sufficiently fast. The variety of knots is

also much higher compared to what has been reported for excitable media [118, 136]. For example,

we have also observed other relevant unknotted structures such as stable double helices (see the

Appendix) — a structure that has remained elusive in the CGLE with local coupling [35]. This

suggests that the models considered here can serve as paradigmatic models to study various knotted

and unknotted structures associated with scroll waves in general, including the novel topological

superstructures of knots with SDSs. More specifically, it allows one to explore the topological

constraints imposed by the phase field on the observable phase twists associated with a given knot

— a field largely untouched [136] — as well as the effect of synchronization defect sheets on knots

for the first time.

A remaining open question is to which extent the existence of stable knots in oscillatory sys-

tems depends on the presence of a chimera state. While our findings suggest that a chimera state

is a necessary condition, there is no fundamental reason to substantiate this. However, our simula-

tions indicate that the mobility of the scroll wave filaments plays an important role. If the filaments

move or meander sufficiently fast (e.g. R = 1 or for a Gaussian kernel with large α), no chimera

state can be numerically observed and stable knots are absent. This is similar to what has been

reported for chimera spirals in 2D [134] and knots in excitable media [118]. One possible way

forward is the recently proposed ansatz by Ott and Antonsen [29, 30], which has been successfully

applied to study the existence and stability of chimera spirals [142].

In addition to the robustness of knots under dynamical noise, we also find that Hopf links

and trefoils can emerge in a self-organized way from random IC with fair probability (see the

Appendix). Both features indicate knots should be observable in real-world oscillatory systems that

follow a dynamics similar to the models studied here, with most likely candidates to be chemical

51

systems [18, 139, 137, 143, 22]. Yet, the observation of chimera filaments in natural systems

remains a challenge for the future.

3.12 Appendix A: Topological structures

Fig. 3.6 shows various long-lived (stable or metastable) topological structures in the non-local

Kuramoto model within the regime αK < α < α0. Note that knotted structures more complicated

than simple Hopf links tend to have smaller stability regimes. An exception are (knotted) structures

that require periodic boundary conditions (BC) and do not drift, which can be stable below αK .

This is shown, for example, in Fig. 3.7 and includes straight filaments. Simulations suggest that

the multi-filament structure in Fig. 3.7c is stable for α > 0.

3.13 Appendix B: Dynamics

In the non-local Kuramoto model, knotted structures that exist independent of the specific choice

of BC (periodic vs. no-flux) are not stationary but drift, rotate and change their shape over time.

As an example, Figs. 3.8(a) and 3.8(b) show a few snapshots for different structures. Note that

for the system sizes studied, the center of mass motion is not straight over long time scales. The

phase field away from these knotted structures takes on the form of spherical waves as shown in

Fig. 3.9. In case of the ring-tube structure (which is specific to periodic BC), the ring propagates

along the tube and keeps distorting the local part of the tube while it travels, see Fig. 3.8 (c). In

all these cases, the direction of the filament motion can be deduced from the instantaneous angular

frequency ω(x,y,z) shown in the rightmost column of Fig. 3.8.

52

(a) τ > 1.2×106, α = 0.8, R = 4 (b) τ > 1.2×106, α = 0.8, R = 4 (c) τ > 250000, α = 0.8, R = 4

(d) τ > 200000, α = 0.7, R = 4 (e) τ > 200000, α = 0.7, R = 4 (f) τ ∼ 70000, α = 0.8, R = 4

(g) τ ∼ 5000, α = 0.8, R = 4 (h) τ ∼ 5000, α = 0.8, R = 4 (i) τ ∼ 5000, α = 0.7, R = 5

Figure 3.6: Non-trivial topological structures in the non-local Kuramoto model with periodic BCfor L = 100. The lifetime τ and the corresponding parameters are given for each subfigure. τ > τ0means that the structure is stable within the testing time limit τ0, while τ ∼ τ0 means the structurebreaks down around τ0 (order of magnitude). The period of the scroll waves is about T ∼ 11 forα = 0.8. This implies a lifetime of more than 105T for Hopf links and trefoils. Together with therobustness in the presence of noise as established in the main text, this suggests that the lifetimeτ → ∞ when L R.

53

3.14 Appendix C: Creating chimera knots

3.14.1 Random initial condition

Knots and links can appear spontaneously from random initial conditions (IC). The transient time

is of the order of one thousand scroll wave periods in the regimes being studied. A few snapshots

of typical transient states are shown in Fig. 3.10. Using random IC, we can obtain all knotted

structures shown in Figs. 3.6a-3.6f. The specific probabilities of generating Hopf links and trefoils

from random IC are summarized in Table 3.1.

3.14.2 Algorithm to create rings and Hopf links

First, the phase field of a single ring is considered. Suppose the center of a ring is located at

r0 = (x0,y0,z0) with radius R0 and the normal vector of the ring is pointing in the positive z

direction. A parameterization of the location of this ring using φ ∈ [0,2π) is

r(φ) = (rx,ry,rz) = (xo +R0 cosφ ,y0 +R0 sinφ ,z0). (3.9)

To create a ring shaped filament corresponding to phase singularities, the phase field needs to

be specified in the whole domain such that it is smooth outside the ring but results in 2π phase

difference while going around a point on the filament. This can be done by defining

ϕ = tan−1(

z− z0

R0− f

)(3.10)

f =√(x− x0)2 +(y− y0)2 (3.11)

for any spatial point r = (x,y,z). Then the phase of each oscillator θ(r) can be computed by

θ(r) = ψ(r), where

ψ(r) = kd−ϕ− sφ −β . (3.12)

Here, k is the wavenumber, d is the distance to the closest point on the ring d =√(x− rx)2 +(y− ry)2 +(z− rz)2,

ϕ is the angle between the plane consisting of the ring and the line to the closest point of the ring,

54

(a) Two simple filaments with notwist (α = 0.8).

(b) Two simple filaments twistingonce (α = 0.8).

(c) Two twisted filaments passingthrough all three surfaces (α = 0.05).

Figure 3.7: Various long-lived filaments with periodic BC, L = 100 and R= 4. (a-b) Each filamentconnects with itself through one of the surfaces. (c) Each filament passes through all three surfacesbefore connecting back to itself.

Table 3.1: Spontaneous formation of Hopf links and trefoils in simulations with random IC andperiodic BC.

α R L Number of simulations Number of Hopf links Number of trefoils0.7 4 100 200 1 10.7 5 100 500 4 00.8 4 100 500 13 00.8 5 100 500 5 00.7 5 200 100 3 10.8 8 200 100 4 0

55

(a) Hopf link

(b) Trefoil

(c) Ring-tube

Figure 3.8: Dynamics of different topological structures (α = 0.8, R = 4 and periodic BC). Thefirst three columns are snapshots at three different instances in time. The rightmost column is theiso-surface plot of the instantaneous angular frequency ω of the last snapshot. Blue indicates theregion with |ω| < |ω|, while orange indicates the region |ω| > |ω|. Filaments are moving awayfrom the orange region.

56

Figure 3.9: The spherical wave generated around a Hopf link is shown by plotting the iso-surfaceθ = 0 (α = 0.8, R = 8 and periodic BC). The irregular pattern at the center is the region withunsynchronized oscillators that form the filament. Note that only the lower half of the system isshown to highlight the structures near the center.

(a) A transient state for R = 8, L = 200.(b) A transient state for R = 4, L = 200.See Sec. 3.14.3 for a discussion of thesignificance of the red box.

(c) Same as in (b) at a later time.

Figure 3.10: Some snapshots of transient states (α = 0.8 and periodic BC).

57

s is the twisting number, φ is the ring parameterization, and β is a constant phase shift. Examples

are shown in Fig. 3.11.

The phase field of a Hopf link can be created by combining two rings, requiring a method to

smoothly superimpose them. This can be achieved using a distance dependent phase:

ξ (r,r0,s,β ) =(

Rd

)2

eiψ(r,s,β ), (3.13)

which is based on the inverse square distance. Then the phase field of a Hopf link θ(r) can be

calculated by

ρ(r)eiθ(r) = ξ (x,y,z,x0−R0/2,y0,z0,s,β = 0)+ξ (x,z,y,x0 +R0/2,y0,z0,s,β = π) (3.14)

with twisting number s = 1. Examples are shown in Fig. 3.12. Note that a structure in a given

system size L can be rescaled to L′ using a simple scaling function of the form θ ′(x′,y′,z′) =

θ(⌊ L

L′ x′⌋ ,⌊ L

L′ y′⌋ ,⌊ L

L′ z′⌋), where b·c denotes the floor of the number (which is necessary since the

oscillators are arranged on a discrete lattice) and the prime denotes the new phase and new location.

This rescaling works quite well for the top-hat kernel as long as R∼ R′ 1. Also, if the smoothed

phase θ of knots — see Eq. (4) in the main text — is used as IC, the unsynchronized region around

the filaments can redevelop.

3.14.3 Reconnecting chimera filaments using random patches

A new structure can be obtained by reconnecting local filaments of a known structure. This re-

connection requires a detailed specification of the whole local phase field that is smooth, without

creating new filaments and while matching the desired filaments. This can be hard to do if the

local filaments are obtained from a simulation. Alternatively, based on the observation that only

simple straight filament can form in a small system size L/R from random IC, it suggests a way to

transform a structure by randomizing a whole local region. Using this method, we have success-

fully created trefoils and a few other knots. To begin with, a structure that is similar to the desired

knot is needed, with the region of reconnection close to each other. For example, the structure in

58

(a)

(b)

Figure 3.11: Shrinking rings with (a) non-local coupling R = 4, (b) nearest-neighbor couplingR = 1. Parameters: L/R = 50, R0/R = 20, α = 0.8 with no-flux BC.

59

(a)

(b)

Figure 3.12: Formation of knots for (a) non-local coupling R = 4, (b) nearest neighbor couplingR = 1. Note that the IC are exactly the same in both cases. Parameters: R0 = 25, L = 100, α = 0.8with no-flux BC.

60

the red box shown in Fig. 3.10 is a trefoil if the top parts are connected. After half a dozen trials

using different shapes of the randomized region, we were indeed able to create a trefoil. Note that

the region should be large enough to form a tube but not too large to form other structures. This

method may suggest a similar way to create knots in real world experiment.

3.15 Appendix D: Instabilities

In the main text, the instabilities at αK and α0 of Hopf links in the Kuramoto model have been

discussed. The instability near αK is caused by a lack of repulsion to counter curvature-driven

shrinkage, so knots collapse and disappear. On the other hand, the instability near α0 originates

from an instability of the filament where the filaments become longer and longer and eventually

collide with themselves or other filaments. This effect is particularly clear in large domains as

shown, for example, in Fig. 3.13. Note that an elongation also happens as a transient state when

the parameters are suddenly changed or starting from a non-perfect IC. However, it will eventually

shorten after refolding to an asymptotic state as also observed for other models [118]. Other

instabilities are discussed below.

3.15.1 Instability of a single ring

Direct simulations show that rings are not stable for α < α0 with no-flux BC. As shown in Fig.

3.11, all rings shrink in size and eventually vanish. The largest ring tested had radius R0 = 80. This

shrinkage process occurs for both nearest neighbor coupling R = 1 and non-local coupling R = 4.

Note that the time it takes for a ring to disappear is approximately the same in both cases for the

same effective radius R0/R and effective system size L/R. Also, almost all transient (knotted)

states resulting eventually in homogeneous oscillations become rings in their penultimate stage.

61

3.15.2 Instability of knots for R = 1

As shown in Fig. 3.12, using the IC for Hopf links described in Section 3.14.2 can result in a

stable knot if R 1. For the choice of R0, the two rings initially shrink in size and then an

effective repulsion prevents further shrinkage. At the same time, the center of the Hopf link starts

moving. In contrast, if the same IC is used with nearest neighbor coupling R = 1, the two rings will

eventually collide with each other and decay into a single ring, which in turn shrinks and vanishes.

3.15.3 Filament instability at α0

As illustrated in Fig. 3.14, the instability at α0 for simple straight chimera filaments is char-

acterized by the emergence of secondary structures and the elongation of filaments. The same

qualitative behavior is observed for knotted structures in the regime α > α0 as shown in the main

text. Nevertheless, the knotted structures can persist for thousands of scroll wave periods before

they break up consistent with critical slowing down near a phase transition.

3.15.4 Instabilities from finite size effects

A stable knotted structure becomes unstable when it is confined in a small effective system L/R.

While a Hopf link simply decays into a single ring which eventually vanishes, the situation is more

complicated for larger and more complex knotted structures. One example is shown in Fig. 3.15

starting from a triple ring for R = 4 in L = 100, which decays into a ring knotted with 8-shape ring,

and then transforms into a trefoil. Depending on the IC and the exact parameter regime, the decay

path can be different. Note that even for the moderately larger system size L = 150, triple rings

have significantly longer lifetimes (τ > 20000) in some parameter regimes.

62

Figure 3.13: Instability of a trefoil near the transition point α0 (R = 4, L = 150 and periodic BC).This snapshot shows the initial elongation of one branch of the trefoil, which has collided withitself and formed an extra ring.

(a)

(b)

Figure 3.14: Snapshot series of the instability of straight filaments at α = 0.95 > α0. (a) Singlefilament with L = 100, R = 4 and no-flux boundary conditions. Some secondary structures developwith local twisting before break-up. (b) Two filaments in a larger domain L = 200 and R = 4 withPBC. The rapid elongation of one of the filaments is evident.

63

Figure 3.15: Decay of a triple ring for L = 100, R = 5, α = 0.7 with PBC. (a) t = 0, triple rings.(b) t = 15000, decay into a ring knotted with an 8-shape ring. (d) t = 20000, further decay into atrefoil.

0 2 4 6 8 10

10−2

10−4

10−6

G0

G1

G2

r

Gi(r)

(b)

Figure 3.16: (color online) Plot of the localized kernels that can still result in a stable Hopf link,with estimated critical values n1 = 3.8 and n2 = 1.9 for R = 4 and α = 0.8 of Kuramoto model.G0 is shown for comparison.

64

3.16 Appendix E: Spatial kernels

As mentioned in the main text, our main findings do not depend qualitatively on the exact func-

tional form of the considered kernels. For example, using the kernel

G′0(r)∼

1, |x|, |y|, |z| ≤ R

0, otherwise(3.15)

instead of the top-hat kernel G0 gives pretty much identical results for the stability of knots. As

another example, using the kernel

G2(r)∼ (1+ en2(r−R))−1 (3.16)

with an exponential tail instead of the kernel G1 with super-exponential tail exhibits the same

phenomenology: With decreasing n2, the stable regime of knots shrinks. The shape of the kernels

at the transition points of G1 and G2 for R = 4 and α = 0.8 are shown in Fig. 3.16.

3.17 Appendix F: Other oscillatory models

3.17.1 Non-Local Complex Ginzburg-Landau equation (CGLE)

The non-local CGLE considered here is [7]:

A(r, t) = A− (1+ ib)|A|2A+K(1+ ia)∫

G(r− r′)(A(r′)−A(r))dr′, (3.17)

where the control parameters are (a,b), the coupling strength is K and G = G0 in the following.

Under sufficiently weak coupling K→ 0, the local field oscillates with unit amplitude |A| ≈ 1 and

behaves like a simple phase oscillator in the non-local Kuramoto model. Therefore, we can use the

knotted structures found in the Kuramoto model as IC by simply setting A(r, t = 0) = eiθ(r). We

find that one of the regimes with stable Hopf links is 0.95 . a . 1.15 for b = 0 and K = 0.1 pro-

vided that L R 1. In Fig. 3.17(a), the phase portrait shows that the magnitude of all oscillators

only deviates slightly from |A| = 1 in this case. For stronger coupling K = 0.2, the deviations in

65

(a) K = 0.1 (b) K = 0.2

Figure 3.17: Snapshot of the states of the oscillators in phase space for a Hopf link in the non-localCGLE for (a,b) = (1,0), L = 200, R = 8 and periodic BC.

A increase (see Fig. 3.17(b)) but stable knots still exist. In both cases, the phase θ(r) = arg(A(r))

behaves similar to the Kuramoto model as confirmed by Fig. 3.18(d). As Figs. 3.18(b) and 3.18(c)

show, the chimera nature is also evident from the Re(A(x,y,z)) and Im(A(x,y,z)) fields. Using the

local mean field θ(r), one can easily locate the unsynchronized filaments 4. An example is shown

in Fig. 3.18(a).

3.17.2 CGLE: Minimum separation & spontaneous fluctuations

When R becomes too small, knots are no longer stable. This instability can be characterized by the

dynamics of the filament(s) that make up the knots. Even though the region around the filament

is unsychronized, the filaments can be found by a filament detection algorithm [131] of the mean

field (see Fig. 3.19(a)). The length of filament can therefore be defined as the number of occupied

lattice sites. Denote the two rings or filaments of a Hopf link as F1 and F2 with circumference

(or length) C1 and C2, respectively. As Fig. 3.19(c) shows, C1 and C2 fluctuate over time in a

synchronous way. Fluctuations are also present in the minimum separation between F1 and F2,

4To identify the regions with unsynchronized phase, we consider the average of the absolute phase difference withits neighbors and select a suitable threshold.

66

(a) Hopf link

(b) Re(A) (c) Im(A) (d) θ = arg(A)

Figure 3.18: Snapshot of a Hopf link in the non-local CGLE corresponding to Fig. 3.17(a). (a)shows the unsynchronized region corresponding to the chimera knot. An x-y cross-section of thedifferent fields at z = 100 is plotted in (b)-(d). In (b) and (c), the color map from deep blue to redcorresponds to values from −1 to 1 in the respective field.

67

(a) Snapshot of the filaments of a Hopf link. (b) Snapshot of a straight filament with no-flux BC.

0 5000 10000 15000 20000

t

80

90

100

110

120

130

140

150

C1,C

2

C1

C2

(c) Circumferences Ci of the two rings shown in (a) asa function of time.

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

t

100

105

110

115

120

125

130

ℓ

(d) Length of the filament ` shown in (b) as a functionof time.

0 5000 10000 15000 20000

t

7

8

9

10

11

12

13

dmin

(e) Minimum separation between the two rings shownin (a) as a function of time.

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

t

1.0

1.5

2.0

2.5

3.0

3.5

∆xy

(f) Roughness of the filament shown in (b) as a functionof time.

Figure 3.19: CGLE with R = 4, a = 1, b = 0, K = 0.1. (left) Temporal evolution of a Hopf linkwith L = 80. (right) Temporal evolution of a single filament oriented along the z-direction withL = 91 which results in a time average filament length 〈`〉 ≈ 110 that is approximately the same asthe time average circumference 〈C〉 ≈ 110 of the rings in the left column.

68

0 1 2 3 4 5 6 7 8

R

0

50

100

150

200

Num

ber

ofla

ttic

esi

te

〈C〉 ≈ 〈ℓ〉Leff

(a) Average circumference and effective system sizeLe f f .

3 4 5 6 7 8

R

0

1

2

3

4

5

6

Num

ber

ofla

ttic

esi

te

dmin,0.995 − dmin,0.005

ξ

(b) Comparing different length scale of fluctuation.

Figure 3.20: CGLE with a = 1, b = 0, K = 0.1 as in Fig. 3.19. (a) System size, Le f f , for whicha single filament has the same average length 〈`〉 as the average circumference 〈C〉 of a Hopf link.(b) Measures of fluctuations for the case of a Hopf link (spread in the minimum separation betweenthe rings, dmin,99.5%−dmin,0.5%, see Fig. 3.19(e)) and for the case of a single filament (spread in theroughness, ξ , see Fig. 3.19(f)), both as a function of R.

69

defined as dmin = minri∈Fi(r1,r2), as shown in Fig. 3.19(e). To characterize these fluctuations

statistically and identify an associated length scale, we consider the difference between the 99.5%-

quantile and the 0.5%-quantile associated with dmin, corresponding to the error bars shown in Fig.

3.5(a) in the main text. As shown in Fig. 3.20(b), this difference is not varying much across the

considered values of R. This is in sharp contrast to the linear scaling of dmin with R (see Fig. 3.5(a)

in the main text).

To substantiate that the intrinsic length scales associated with filament fluctuations do not

strongly vary with R, we further consider the fluctuations of a single straight filament (see Fig. 3.19(b)).

To ensure a fair comparison with the fluctuations of Hopf links, we choose a system size L = Le f f

such that the average single filament length 〈`〉 equals the average circumference 〈C〉 = (〈C1〉+

〈C2〉)/2 of the filaments in the Hopf link (see Fig. 3.19(d)). The dependence of both these quanti-

ties as a function of R is shown in Fig. 3.20(a). To characterize the fluctuations of a single straight

filament, we calculate its roughness. Due to the chosen initial conditions, the roughness is identi-

cal to the deviation from a straight filament oriented along the z-axis. Specifically, we define the

deviation from the straight filament center rxy = (1/L)∑z rxy(z) to be

∆xy(z) = |rxy(z)− rxy|, (3.18)

where rxy(z) is the intersection point of the filament with the x-y plane for a given z. The roughness

∆xy = (1/L)∑z ∆xy(z) is now simply ∆xy(z) averaged over z. As Fig. 3.19(f) shows, the roughness

varies over time. To characterize these (non-negative) fluctuations in the roughness over time and

within an ensemble and to identify an associated length scale, we consider the 99%-quantile and

denote it by ξ . This is the quantity shown in Fig. 3.5(a) in the main text and again it does not

vary much across the considered values of R. For a direct comparison with the length scale of

fluctuations in the case of a Hopf link, please see Fig. 3.20(b).

70

3.17.3 Non-Local Rossler model

The non-local Rossler model considered here is [139]:

X(r, t) = −Y −Z +K∫

G(r− r′)(X(r′)−X(r)

)dr′, (3.19)

Y (r, t) = X +aY +K∫

G(r− r′)(Y (r′)−Y (r)

)dr′, (3.20)

Z(r, t) = b+Z(X− c), (3.21)

where the control parameters are (a,b,c) and the coupling strength is K. Again, we can use

(X ,Y,Z) = (cosθ ,sinθ ,0) with θ(r) from states with knotted structures generated by the Ku-

ramoto model as IC. When a = b = 0.2, the effective |α| decreases as c increases [138]. We

observe stable knots within 3.3 . c . 5 for weak coupling K = 0.05 provided that L R 1.

Note that as c increases the intrinsic dynamics of the oscillators also changes. Namely, the dynam-

ics undergoes a period-doubling cascade to chaotic oscillations. In particular, we observe stable

knots in the period-2 regime with c = 3.6 (Fig. 3.21), as well as in the chaotic regime with c = 4.8

(Fig. 3.22).

An additional feature of the wave dynamics in these regimes is evident from Fig. 3.21 and

Fig. 3.22: The amplitudes are modulated. For example, in the period-2 regime alternating wave

maxima are present. A topological consequence of such a behavior is that two dimensional struc-

tures exist such that the local dynamics has a lower period than that of the bulk. Specifically,

these structures, called synchronization defect sheets (SDSs) in the following, separate domains

of different oscillation phases and for periodic BC either originate from a filament or are closed.

More importantly, every filament has an attached SDS such that they become part of any knotted

structure. This can already be observed in the cross-sections shown in Fig. 3.21 and Fig. 3.22. To

clearly identify SDSs, we use the detection algorithm developed for the lower dimensional case

in Ref. [138]. A specific example of SDSs is shown in Fig. 3.23 and their subsequent motion is

shown in the Supplementary Video.

71

(a) Hopf link

(b) X (c) Y

(d) Z (e) θ = tan−1(Y/X)

Figure 3.21: Snapshot of a Hopf link in the non-local Rossler model for(a,b,c,K) = (0.2,0.2,3.6,0.05), corresponding to the period-2 regime. The lifetime of theknot is τ > 105. The same 2D cross-sections of the Hopf link are shown in (b-e) for the differentfields X , Y , Z and θ . The color scheme is such that deep blue represents the most negative value,and red represents the most positive value. The discontinuities of color along the wave frontscorrespond to cross-sections of synchronization defect sheets. L = 200, R = 8 and periodic BC.

72

(a) Hopf link

(b) X (c) Y

(d) Z (e) θ = tan−1(Y/X)

Figure 3.22: Similar to Fig. 3.21, but in the chaotic regime (a,b,c,K) = (0.2,0.2,4.8,0.05).

73

(a) Synchronization defect sheets (b) Synchronization defect sheets (another view)

(c) z = 50 (d) z = 85 (e) z = 93

(f) z = 109 (g) z = 118 (h) z = 130

Figure 3.23: Visualization of the synchronization defect sheets (SDSs) present in Fig. 3.21. (a),(b): Different 3D plots of the SDSs. (c)-(h): 2D cross-sections at different values of z. The redlines represent the cross-sections of SDSs and the yellow-red dots indicate the unsynchronizedregions. Note that the cross-section in (e) is the same cross-section as in Fig. 3.21 (b-e).

74

Chapter 4

Proposal for the Creation and Optical Detection of Spin Cat

States in Bose-Einstein Condensates

4.1 Preface

We propose a method to create “spin cat states”, i.e. macroscopic superpositions of coherent spin

states, in Bose-Einstein condensates using the Kerr nonlinearity due to atomic collisions. Based on

a detailed study of atom loss, we conclude that cat sizes of hundreds of atoms should be realistic.

The existence of the spin cat states can be demonstrated by optical readout. Our analysis also

includes the effects of higher-order nonlinearities, atom number fluctuations, and limited readout

efficiency.

The work in this chapter was published in [12]. The detailed calculations in this chapter were

the results of my own work. The original idea was proposed by Prof. Simon. Under his guidance,

I was able to work out all the details of this proposal. During the development of this proposal,

Dr. Zachary Dutton provided valuable suggestions thanks to his expertise in BECs. Another BEC

expert I consulted is Dr. Rui Zhang. Their involvement made my proposal more realistic. Tian

Wang was involved in the part related to the Kerr effect.

4.2 Introduction

Great efforts are currently made in many areas to bring quantum effects such as superposition and

entanglement to the macroscopic level [144, 145, 146, 147, 148, 149, 150, 151, 97, 152, 153, 154,

155, 156, 157, 51]. A particularly dramatic class of macroscopic superposition states are so-called

cat states, i.e. superpositions of coherent states where the distance between the two components

in phase space can be much greater than their individual size [144, 145]. For example, the recent

75

experiment of [51] created a cat state of over one hundred microwave photons in a waveguide

cavity coupled to a superconducting qubit. It was essential for the success of the latter experiment

that the loss in that system is extremely small, since even the loss of a single particle from a cat

state of this size will lead to almost complete decoherence.

Here we show that it should be possible to create cat states involving the spins of hundreds of

atoms in another system where particle losses can be greatly suppressed, namely, Bose-Einstein

condensates (BECs), where the spins correspond to different hyperfine states. We use the Kerr

nonlinearity due to atomic collisions, which also played a key role in recent demonstrations of

atomic spin squeezing [97, 152, 153]. In contrast to previous proposals [107, 158] we do not make

use of Josephson couplings to create the cat state, but rely purely on the Kerr nonlinearity in the

spirit of the well-known optical proposal of Ref. [59].

Our approach is inspired by the experiment of Ref. [159], which stored light in a BEC for

over a second. Ref. [160] proposed to use collision-based interactions in this system to implement

photon-photon gates, see also Ref. [161]. Here we apply a similar approach to the creation and

optical detection of spin cat states. Because of the great sensitivity of these states, this requires a

careful analysis of atom loss. Our theoretical treatment goes beyond that of Ref. [160], which was

based on the Thomas-Fermi approximation (TFA). Our new approach allows us to study several

key imperfections in addition to loss, including higher-order nonlinearities, atom number fluctua-

tions, and inefficient readout, and we conclude that their effects should be manageable.

4.3 Spin cat states creation scheme

Our scheme is illustrated in Fig. 4.1. The setup is similar to the experiment of Ref. [159]; See also

Ref. [164]. In particular, the light is converted into atomic coherences using a control beam (’slow’

and ’stopped’ light) [165, 159, 166, 167, 168, 169, 170]. We start with a ground state BEC with

N atoms in internal states |A〉. To create a spin state, a coherent light pulse, |α〉L = ∑n cn |n〉 with

mean photon number n = |α|2 and cn = e−|α|2/2(αn/

√n!), is sent into the BEC (see Fig. 4.1a).

76

(a)

zBEC

Light(b)

zAB

(c) t = 0(CSS)

z

ωa

ωb(d) t = τc

(CAT)

z

(e)

zAB

(f)

zBEC

HomodyneDetectors

t = 0 CSS t = τc CAT t = 2τc CSS

|A〉 |B〉

|C〉

probebeam

couplingbeam

(g)

Im(β)

Re(β)

Figure 4.1: (color online) Spin cat state creation (a)-(d) and detection (e)-(g). In (a)-(f) theradially symmetric photons and spherically symmetric BECs are represented by spatial densitydistributions. (a) A coherent light pulse is sent into the BEC. (b) The light state is absorbed in theBEC (see inset), creating a CSS. The shape of the input pulse is chosen such that the two-compo-nent BEC is in its ground state after the absorption. (c) The trapping frequency ωb for the smallcomponent is increased adiabatically. The density of the small component now exceeds that of thelarge component at the center. (d) The collision-induced Kerr nonlinearity drives the system into aspin cat state (CAT). (e) The trapping frequency is adiabatically reduced to its initial value. (f) Thespin state is reconverted into light, whose Husimi Q function [47] is determined via homodyne de-tection [162]. (g) Expected shape of the Q(β ) function in phase space. The coherent state at t = 0gives a single peak, while the cat state at t = τc yields two peaks. Further evolution for anotherinterval τc returns the output light to a coherent state, yielding a single peak at t = 2τc. This wouldnot be possible if the two peaks at t = τc corresponded to an incoherent mixture, thus proving theexistence of a coherent superposition of CSSs in the BEC at τc [163].

The light is absorbed by the BEC and some atoms are converted into internal states |B〉 as:

∑n

cn |n〉L |N,0〉S→ |0〉L ∑n

cn|N−n,n〉S := |0〉L|α〉S, (4.1)

where the Fock state |Na,Nb〉S represents Na and Nb excitations of wavefunctions ψa and ψb in the

A and B components respectively. Note that |α〉S is an excellent approximation of a CSS [171]

∑Nn=0

√N!/(n!(N−n)!)αn|N−n,n〉 in the limit of N n which is the case in this scheme.

The described absorption process should prepare the two-component BEC in its motional

ground state to avoid the complication of unnecessary dynamics such as oscillations. This can

77

be achieved by matching the shape of the input pulse to the ground state of the effective trapping

potential for the small component [160], provided that the effective trap is not too steep. Once the

light has been absorbed, the trapping frequency ωb is then increased adiabatically independently of

ωa, which can be achieved by combining optical and magnetic trapping. In the regime ωb ωa,

a narrow wavefunction ψb is formed at the center and its density can exceed the large component

A, see Fig. 4.1c. This results in strong self-interaction and hence a large Kerr nonlinearity. On the

other hand, keeping ωa low reduces the unwanted effects due to collision loss involving the large

component.

The spin state will now evolve with time according to

|χ(t)〉S = ∑n

cne−iE(N,n)t/h|N−n,n〉S (4.2)

with |χ(0)〉S = |α〉S. If the energy takes the Kerr nonlinear form H = hη2n2, then a spin cat

state |χ(τc)〉S = (|α〉S + i|−α〉S)/√

2 is formed at the time τc = π/|2η2| in full analogy with the

proposal of Ref. [59]. The problem is thus reduced to the computation of the ground state energy

E(N,n).

4.4 Calculating energy

The energy of the system can be calculated by the following mean-field energy functional E[ψa,ψb;Na,Nb]:

E = ∑i=a,b

Ni

(Ki +Vi +

12(Ni−1)Uii

)+NaNbUab (4.3)

where Ki, Vi, Uii and Uab are the kinetic energy, potential energy, intra- and inter-component in-

teraction energy respectively, given by Ki =∫(h2/2m)|∇ψi|2, Vi =

∫Vi|ψi|2 with spherically sym-

metric trapping Vi =mω2i r2/2, and Ui j =

∫Ui j|ψi|2|ψ j|2 with interaction strength Ui j = 4π h2ai j/m.

Here, ψi are single particle wavefunctions for i-th component with normalization∫ |ψi|2d3r = 1, ai j

are the scattering lengths, and m is the atom mass. The corresponding dynamic equation governing

the system evolution is the Gross-Pitaevskii equation (GPE) [172, 173, 10]. With the restriction

78

Na = N−n and Nb = n of spin states creation in Eq. (4.1), the nonlinearity in n can be obtained by

the expansion of the energy E(N,n) = hη(N,n) around n = 0 as:

η(N,n) = η0(N)+η1(N)n+η2(N)n2 +η3(N)n3 + ... (4.4)

where hη0 generates a global phase and hη1 = −µa + µb with chemical potential µi (i = a,b) is

the energy to remove one atom from |A〉 and add one atom to |B〉. hη1 generates a simple rotation

in phase space |α〉 → |αe−iη1t〉, which can be eliminated by a frame rotation. The term η2 is the

Kerr nonlinearity. We obtain these coefficients by fitting the total energy E(N,n) with n ∈ [0,200]

up to fourth orders in Eq. (4.4), where the numerical ground state ψi of GPE used in Eq. (4.3)

is found by the imaginary time method [174]. This numerical approach is better than Ref. [160]

because we can avoid the problems associated with the TFA of high densities [175]. Also, the high

density for the small component at the center limits the negative effect of quantum fluctuations in

the large component [176]. The latter are less important than the classical fluctuations in ηk(N)

due to uncertainty in N, whose effects will be discussed below. Moreover, we can now study the

effects of higher-order nonlinearities (in particular η3 and η4).

4.5 Cat creation time

Fig. 4.2 shows our results for the spin cat creation time τc = π/|2η2| and achievable cat size n,

taking into account the effects of atom loss. It is clear that the cat time τc decreases significantly as

the trapping strength ωb increases. Note that the Kerr effect disappears (η2 = 0) around ωb ≈ 2π×

55Hz, which may be used for long term storage. As mentioned above, the reason for the strong

Kerr effect for large ωb is that strong trapping potential forces ψb into a highly localized Gaussian

φ0(r) = (mωbπ h )3/4e−(mωbr2)/2h. The radius of ψb is of the order of the characteristic length sb =

√π h/(mωb), and the density ρb(r) = n|ψb(r)|2 is peaked at the center ρb(0) ≈ ns−3

b which can

be much higher than ρa(0) in our regime, see Fig. 4.1d. Therefore, the system can be effectively

described by H ≈ 12Ubbn(n−1)

∫d3r|φ0|4, and the second order term is approximately hη2(N)≈

79

0.01

0.1

1

10

100

1000

τ3,baaτ3,bbaτ3,bbb

τ1,bτℓτc

50 100 200 500 1000 2000 4000ωb/2π(Hz)

1

10

100

400

n

N = 104, τc = τℓN = 105, τc = τℓN = 105, τc = 0.1τℓ

τ(s)

τc > τℓ

τc < τℓ

(a)

(b)

Figure 4.2: (color online) (a) The time to create a spin cat state τc = π/|2η2| (thick red curve)versus the time to lose one atom τ` in component B (thick black curve) as a function of the trappingfrequency for the small component ωb. The size of the cat n = 100 in this example. One sees thatτc < τ` is possible for sufficiently large ωb. The plot also shows the main individual loss channelscontributing to the calculation of τ`, where τm,c is the individual time of losing one atom throughm-body collision with particle combinations c. It furthermore shows analytic approximations forτc ∼ ω

−3/2b (red dotted curve) and τ3,bbb ∼ ω

−3b (blue dashed curve), see text. (b) Achievable cat

size n as a function of ωb. The shaded region corresponds to τc < τ` for a condensate size N = 105

as in (a). The cat size can be increased somewhat by reducing N (dashed line). We also showthat there is a region where τc < 0.1τ` so that loss should really be negligible. The green circlescorrespond to τc = 10,1,0.1s (from left to right). The star corresponds to the values used in Fig.3, and the corresponding density distributions are shown in Fig. 1(d). Both plots are for 23Nawith spin states |A〉 = |F = 1,m = 0〉, |B〉 = |F = 2,m = −2〉, scattering lengths aaa = 2.8nm,abb = aab = 3.4nm [159], loss coefficients L1 = 0.01/s, L2 = 0, L3 = 2×10−42m6/s [177], and atrapping frequency ωa = 2π×20Hz for the large component.

80

(Ubb/2)∫

d3r|φ0|4 = Ubb2−5/2s−3b , which is consistent with the first order perturbation theory in

the Appendix.

4.6 Atom loss

The phase between the two components of the spin cat state is flipped by losing just one atom (see

the Appendix for more details on the effects of atom loss). This means that τc must be smaller than

the time to lose one atom τ`, which depends on the density and thus n. In our scheme, the loss of

atoms in component A will not affect the cat states directly, so we focus on the loss of component

B only, which can be estimated by the following loss rate equation [69, 178, 179]:

dn/dt =−τ−1` =−(L1 +L2 +L3) (4.5)

where τ` = 1/(L1 +L2 +L3) is the approximate time to lose one atom through all possible loss

channels if n 1. The loss rates Lm correspond to the loss through m-body collisions involving

particles in component B, where L1 = L1n is due to collisions with the background gas, L2 =

∑ j L2,b j∫

ρbρ j is due to spin exchange collisions, and L3 = ∑ j,k L3∫

ρbρ jρk is due to three-body

recombination [69]. It is known that the two-particle loss can be eliminated by certain choices

of internal states and control methods such as applying a microwave field in [96], or a specific

magnetic field as in Ref. [159]. The latter example motivates our choice of parameters in Fig. 4.2.

Fig. 4.2a shows the time to lose one atom through different channels: τ1 = (L1n)−1 for one-

body loss and τ3,i jk = (L3∫

d3rρiρ jρk)−1 for three-body loss with different combination of colli-

sions. It can be observed that the high ωb regime is dominated by the loss of∫

ρ3b ∼ s−6

b n3, which

corresponds to τ3,bbb. For even larger values of ωb than those shown in the figure, the three-body

loss time τ3,bbb becomes shorter than τc. The small ωb regime is dominated by the effect of τ1.

See the Appendix for an approximate analytical treatment of atom loss. The desirable region for

experiments is τc < τ` which also depends on n. Therefore, we can draw a n-ωb phase diagram,

which shows the achievable cat size as the shaded area in Fig. 4.2b.

81

4.7 Detection scheme

We now discuss how the existence of the spin cat states can be demonstrated via optical readout,

see also Fig. 1(e) to 1(g). Our detection scheme is based on a revival argument and hence involves

measurements at different times [163] (See also the related experiment of Ref. [180]). In all cases

the readout process starts by reducing the trapping frequency adiabatically to its initial value. Then

the spin state |χ(t)〉S is reconverted into a state of light |χ(t)〉L, followed by homodyne detection

on the output light. Using optical homodyne tomography [162], we can reconstruct the Husimi

Q-function [47] Q(β , t) = 1π〈β |ρ(t)|β 〉 with the density matrix ρ(t) = |χ(t)〉L 〈χ(t)|. The Q-

function allows us to visualize the resulting spin states of BEC as a function of time.

Higher-order nonlinearities distort the cat state and shift the cat creation time from τc for a pure

Kerr nonlinearity to a different observed value τ∗c . Fig. 4.3(a) shows Q(β ,τ∗c ) for ωb = 2π×500Hz

including up to fourth-order nonlinear terms ηk. Two peaks at t = τ∗c can be identified clearly. At

the revival time t = 2τ∗c , a single peak is recovered, which proves the existence of spin cat states in

the BEC at τ∗c , as described in Fig. 4.1(g). Note that the definition of τ∗c used is the time at which the

Q-function shows the two highest peaks. In general, η3 < 0 and hence τ∗c > τc for ωb ωa since

ψb is less localized than φ0 due to the repulsive self-interaction. For the weakly phase separated

regime (aaaabb . a2ab) used in Fig. 4.2, the effective compression from component A on ψb can

have the reverse effect. This gives η3 ≈ 0 and thus nearly perfect cat states at ωb ≈ 2π×400 Hz.

Further higher-order effects are shown in the Appendix.

In current experiments the light storage and retrieval process involves significant photon loss,

e.g. about 90% loss in Ref. [159]. Its main effect is to move the peaks towards the origin, see

Fig. 3b and Appendix. One important requirement for achieving high absorption and emission

efficiency is high optical depth. For the example of Fig. 4.2, the optical depth can be estimated as

d ∼ Nλ 2/(πR2) = 34 with N = 105, wavelength λ = 590nm and the BEC radius R = 18µm. This

is in principle sufficient to achieve an overall efficiency close to 1 [181].

Another important experimental imperfection is the fact that the total atom number N cannot

82

−10 −5 0 5 10

−10

−5

0

5

10

−10 −5 0 5 10

−10

−5

0

5

10

0.00

0.05

0.10

0.15

0.20

−4 −2 0 2 4

−4

−2

0

2

4

−4 −2 0 2 4

−4

−2

0

2

4

0.00

0.02

0.04

0.06

0.08

t = τ ∗c t = 2τ ∗c Q(β)

(a)

(b)

Re(β)

Im(β)

Figure 4.3: (color online) Optical demonstration of the spin cat state in the presence of vari-ous imperfections for the parameter values corresponding to the star in Fig. 2(b) (n = 100 andωb = 2π×500Hz). The spin state is reconverted into light and the Husimi phase space distributionfunction Q(β ) is determined via homodyne tomography. (a) Includes the effects of the higher-ordernonlinearities η3 and η4. Two far separated peaks corresponding to the cat state are clearly visibleat τ∗c = 0.68s, and one peak corresponding to the revived coherent state at 2τ∗c . The shift of thecat creation time due to the higher-order terms is τ∗c /τc = 1.06. (b) Furthermore includes 90%photon retrieval loss, which moves the peaks towards the origin, and 5% uncertainty in the totalatom number, which spreads the peaks in the angular direction.

83

be precisely controlled from shot to shot. This leads to fluctuations in the nonlinear coefficients ηk.

The most important negative effect of these fluctuations is dephasing, i.e. angular spreading of the

peaks in Fig. 3 in phase space [46]. The magnitude of the angular spread at the time τc = π/|2η2|

of the cat state creation can be estimated as ∆ϕ = π∆N2η2

∑k knk−1 ∂ηk∂N , where ∆N is the uncertainty in

N, as discussed in more detail in the Appendix. We find that the sensitivity of our scheme to atom

number fluctuations is minimized for ωb ≈ 2π × 600Hz. Fig. 3b shows that a 5% uncertainty in

N can be tolerated for ωb = 2π×500 Hz (even when occurring in combination with 90 % photon

loss).

4.8 Summary

Two key ingredients for the success of the present scheme are the use of a high trapping frequency

for the small component and the achievement of very low loss. The high trapping frequency

enhances the strength of the Kerr nonlinearity, making it possible to create cat states without relying

on a Feshbach resonance as proposed in Ref. [160]. This makes it possible to avoid the substantial

atom loss typically associated with these resonances [69], and also allows one to use the magnetic

field to eliminate two-body loss, which is critical. For example, the loss rates for the choice of

Rubidium internal states discussed in Ref. [179] would only allow cat sizes of order ten atoms,

see the Appendix. The high trapping frequency also helped us to suppress the unwanted effects

of higher-order nonlinearities and atom number fluctuations. If the readout efficiency could be

increased significantly, then the present scheme could also be used to create optical cat states.

Besides their fundamental interest, both spin cat states and optical cat states are attractive in the

context of quantum metrology [182].

4.9 Appendix A: Properties of two-component BEC

The most important results in the main text and these Appendixes are based on numerical methods.

Therefore, the results can be considered exact within the domain of validity of the equations we

84

25 50 100 200 400 600

ωb/2π(Hz)

1

2

4

8

16

FWHM(µm)

FWHM

FWHMapprox

(a) Full width at half maximum(FWHM) for density ρb

0 100 200 300 400 500 600

ωb/2π(Hz)

0

5

10

15

20

25

ρ(µm

−3)

ρa(0)

ρb(0)

ρb(0)approx

(b) Density distribution at center forboth components

0 100 200 300 400 500 600

ωb/2π(Hz)

0

1000

2000

3000

4000

5000

6000

η 1(H

z)

η1

η1,approx

(c) Linear coefficient η1 and its ap-proximation Eq. (4.21)

30 40 50 60 70 80 90 100

ωb/2π(Hz)

−0.01

0.00

0.01

0.02

0.03

0.04

0.05

η 2(H

z)

η2

η2,approx

102 103 10410−2

10−1

100

101

102

(d) Kerr coefficient η2 and its approx-imation Eq. (4.23)

0 100 200 300 400 500 600

ωb/2π(Hz)

−0.0010

−0.0008

−0.0006

−0.0004

−0.0002

0.0000

η 3(H

z)

(e) Third order coefficient η3

0 200 400 600 800 1000 1200 1400 1600

ωb/2π(Hz)

0.9

1.0

1.1

1.2

1.3

τ∗ c/τ

c(s)

τ ∗c (n = 16)

τ ∗c (n = 64)

τ ∗c (n = 100)

(f) The shift of the ‘best’ cat timeτ∗c /τc

Figure 4.4: Properties of spin states in the two-component BEC for the scheme with cat sizen = 100. (a) The width of component B becomes close to the width of a Gaussian as in Eq.(4.9) around ωb ≈ 2π50Hz (red dash curve). The deviation at high ωb is because of self-repulsionin component B. Also, in this weakly phase separated regime aaaabb . a2

ab with equal trappingωa = ωb = 2π20Hz, the component B is located outside of component A. The component B onlypeaks at the center with ωb about 10% higher than ωa. (b) Density ρa(r = 0) and ρb(r = 0) at thecenter of the trap. Note that the density ρb(0) becomes greater than ρa(0) around ωb/2π ≈ 250Hz(see Fig. 4.5 for a spatial distribution). This suggests that most effects from the main BEC compo-nent A, including its quantum depletion, should be relatively small beyond ωb/2π > 250Hz. Thered dashed curve is the density of the Gaussian approximation Eq. (4.9) (c) The numerical resultsfor η1 show a good agreement with first order perturbation theory. (d) The numerical solution forη2 crosses zero around ωb/2π ≈ 55Hz, which causes the cat time τc = π/|2η2| to diverge aroundthis point. The inset shows that the numerical results approach the simple scaling η2 ∼ ω

3/2b at

large ωb. (e) The third order term η3 also shows a zero-crossing point at around ωb/2π ≈ 375Hz,which is a good region to observe nearly perfect cat states with small n. (f) The relative change ofthe best real cat time τ∗c from τc = π/|2η2|. The region with τ∗c /τc > 1 is roughly ωb/2π & 375Hzdepending on n, which corresponds roughly to the region η3 < 0 in subfigure e, and vice versa.Note that the fourth order term is included when determining τ∗c , see text for its definition and Fig.4.6. The parameters used here are the same as in Fig. 2 in the main text: 23Na with spin states|A〉= |F = 1,m = 0〉, |B〉= |F = 2,m =−2〉, scattering lengths aaa = 2.8nm, abb = aab = 3.4nm[159], loss coefficients L1 = 0.01/s, L2 = 0, L3 = 2×10−42m6/s [177], and a trapping frequencyωa = 2π×20Hz for the large component.

85

used, without relying on analytic approximations. The two-component BEC can be described by

the mean-field Gross-Pitaevskii equation (GPE) [172, 173, 10]. However, the typical analytical

treatment, the Thomas-Fermi approximation (TFA), [183] which ignores the kinetic energy term,

is not reliable in our case. It is known that TFA cannot be used in the case of high density [175],

which is the case we are studying. Instead, we numerically solve the GPE:

ih∂

∂ tψi =

[− h2

2m∇

2 +Vi + ∑j=a,b

Ui j(Ni−δi j)|ψ j|2]

ψi (4.6)

where δi j is the Kronecker delta which cannot be ignored if Ni is of order one; ψi and Ni are the sin-

gle mode wavefunction and the number of particles of the i-th BEC component respectively. The

normalization is∫

d3r|ψi|2 = 1 and the density is given by ρ(r) = Ni|ψi(r)|2. The trapping poten-

tial is Vi = mω2i r2/2, with trapping strength ωi, and the interaction strength is Ui j = 4π h2ai j/m,

with scattering length ai j between component i and j. Our target is to find the ground state energy

and wavefunction, which can be done by using the imaginary time method [174]. First, we use

a Wick rotation t → −it on Eq. (4.6) to obtain the corresponding diffusion equation, which is

then reduced to two coupled 1D non-linear diffusion equations with the assumption of spherical

symmetry. Finally, we let the system relax to the ground state with the fourth order Runge-Kutta

method in time and finite difference method in space. After finding the ground state wavefunction,

we can use it to calculate the mean-field energy functional:

E[ψa,ψb;Na,Nb] = ∑i=a,b

Ni

∫d3r(

h2

2m|∇ψi|2 +Vi|ψi|2 +

12(Ni−1)Uii|ψi|4

)+NaNb

∫d3rUab|ψa|2|ψb|2

(4.7)

which depends on the spatial mode ψi and the number of particles Ni. Note that the spatial modes

ψi depend implicitly on Ni through Eq. (4.6). In our scheme, the focus is the ground state energy

E(N,n) as a function of Na = N−n and Nb = n because the total number of particles N = Na+Nb

in the two-component BEC is fixed. After solving a set of BECs with different small component

in the range n ∈ [0,200], we fit the results up to fourth order to get the expansion coefficients

1h

E(N,n) = η(N,n) = η0(N)+η1(N)n+η2(N)n2 +η3(N)n3 +η4(N)n4 + ... (4.8)

86

Fig. 4.4 shows how the most relevant properties of the ground state of the two-component

BEC change with ωb. For the scheme described in the main text, the interesting regime is when

component B is located at the center of the trap. This can be achieved with a slightly higher

trapping for ωb in this weakly phase separated regime as described in Fig. 4.4 with cat size n= 100.

Note that in the case of equal trapping ωa = ωb, the small component B will locate outside of

component A because of the effective repulsion in this regime. As shown in Fig. 4.4a, the width

of ρb is close to the width of a Gaussian at around ωb/2π = 50Hz, while at higher ωb, the width is

larger than the corresponding Gaussian because of the self-repulsion with other atoms in the same

component B. The same effects can be observed for the real density ρb(r = 0) at the center (Fig.

4.4b), which is lower than the corresponding Gaussian density with ωb. When ωb/2π 250Hz,

the component B has higher density than the main component A. This allows us to ignore most

effects of the component A, including the quantum depletion. Fig. 4.4c-e shows the expansion

coefficients ηk. Note that both η2 and η3 have zero-crossing points. With zero Kerr coefficient,

η2 = 0, the system may be used to store spin states for a long time. Also, the zero third order,

η3 = 0, suggests a regime to create good small spin cat states. Fig. 4.4f shows the effects of the

third order term on the shift of the “best” cat time τ∗c , see definition below.

Qualitatively, the change in η2 with respect to ωb can be understood as follow. The contribu-

tions to the Kerr nonlinearity come from intra-species (aa, bb) and inter-species (ab) interactions,

which have opposite sign to each other. When the trapping is weak and identical for both compo-

nents, the Kerr nonlinearity is close to zero. Also, for the phase separated regime, the component

B is staying in the outer region. When the trapping frequency ωb for the B component is increased,

the B component moves to the center and the overlap between A and B increases at first, which

leads to an increase in the inter-species interaction term, resulting in a larger and negative Kerr

nonlinearity. For very strong trapping of the B component, the overlap between A and B decreases

again whereas the intra-species interaction for the B component increases strongly, leading to a

large positive Kerr nonlinearity. This explains the crossover from negative to positive Kerr non-

87

0 5 10 15 20

r(µm)

0

5

10

15

20ρ(µm

−3)

ρa(r): Numerical density for A

ρb(r): Numerical density for B

ρa0(r) = (µa − Va)/Uaa

ρb0(r) = (mωb

πh )3/4e−mωbr2/(2h)

ρa0(r) = (µa − Va − Uabρb0)/Uaa

Figure 4.5: Numerical density distribution for both components (A and B) and its approximationwith cat size n = 100 and trapping strength ωb/2π = 500Hz. ρa0(r) and ρb0(r) are the unperturbedwavefunction used by the first order perturbation calculation. ˜ρa0(r) is another approximation. Seetext for details. Other parameters used are the same as in Fig. 4.4.

linearity as shown in Fig. S1d. In contrast, for the non-phase separated regime, the B component

always stays inside the A component, and there is no crossover as discussed in Section IV (see Fig.

S5).

4.10 Appendix B: Ground state energy from first order perturbation theory

The numerically obtained spatial density distribution ρi is shown in Fig. 4.5. The approximate

solution of a harmonic oscillator ground state ρb0 for B is good. If we follow a Thomas-Fermi

approach similar to the one used in the previous paper [160] by dropping the kinetic energy term

in Eq. (4.6), we will get ρa0 = (µa−Va−Uabρb0)/Uaa. As expected, this approximation is not

good and gives a negative density as shown in Fig. 4.5. In contrast, the TFA solution for a single

component BEC ρa0 gives a fair approximation for A, given by [173, 10]:

φa0(r;Na) =

√µa0(Na)−Va

NaUaa,

φb0(r) =(mωb

π h

)3/4e−mωbr2/(2h),

µa0(Na) =12

hωa

(15aa√

h/(mωa)

)2/5

N2/5a

µb0 =32

hωb

(4.9)

88

Therefore we perform first order perturbation theory with the following splitting for the GPE:

ih∂

∂ tψa = (− h2

2m∇

2 +Va +NaUaa|φa|2︸ ︷︷ ︸

Ha0

+NbUab|φb|2︸ ︷︷ ︸Ha1

)ψa (4.10)

ih∂

∂ tψb = (− h2

2m∇

2 +Vb︸ ︷︷ ︸

Hb0

+NaUab|φa|2 +(Nb−1)Ubb|φb|2︸ ︷︷ ︸Hb1

)ψb (4.11)

where Hi0 is the unperturbed Hamiltonian and the perturbation is given by Hi1. Note that Na−1≈

Na is used. The solutions of Hi0 are given by Eq. (4.9).

To calculate the energy analytically, we expand the ground state energy E(Na,Nb) as the Taylor

series:

E(Na,Nb) = E(Na, Nb)+ ∑i=a,b

∂E∂Ni

∣∣∣∣(Na,Nb)

(Ni− Ni)+12 ∑

j=a,b∑

i=a,b

∂

∂N j

∂E∂Ni

∣∣∣∣(Na,Nb)

(Ni− Ni)(N j− N j)+ ...(4.12)

Note that the chemical potentials (energy change with respect to the number of particles) are given

by µi(Na,Nb) =∂E∂Ni

(Na,Nb). Since the main component A in the scheme is much larger than the

small component B, or N−n n, the expansion can be carried out around the point (N,0) :

hη0(N) = E(N,0) (4.13)

hη1(N) = −µa(N,0)+µb(N,0) (4.14)

hη2(N) =12[∂Na µa(N,0)−∂Nb µa(N,0)−∂Na µb(N,0)+∂Nb µb(N,0)] (4.15)

Note that the µi here denote the exact chemical potentials from the GPE, which can be approxi-

mated by the unperturbed µi0 plus the perturbed chemical potential ∆µi:

µi = µi0 +∆µi (4.16)

Using the unperturbed solutions Eq. (4.9), the chemical potential can be calculated as:

∆µa = UabNb⟨φa0∣∣|φb0|2

∣∣φa0⟩

(4.17)

=Nb

Na

(Uab

Uaaµa0(Na)−

34

Uabωa

Uaaωbhωa

)(4.18)

∆µb = UabNa⟨φb0∣∣|φa0|2

∣∣φb0⟩+Ubb(Nb−1)

⟨φb0∣∣|φb0|2

∣∣φb0⟩

(4.19)

=

(Uab

Uaaµa0(Na)−

34

Uabωa

Uaaωbhωa

)+Ubb(Nb−1)(

√2sb)

−3 (4.20)

89

where si =√

π h/(mωi) is the characteristic length of a Gaussian. Note that the perturbation

involves an integration whose range is chosen to be the whole space for simplicity, which is justified

by the fact that component B is much narrower than component A when ωb ωa (see Fig. 4.5).

Substituting these results back into η1 in Eq. (4.14), we have:

hη1(N) =−µa0(N)+32

hωb︸ ︷︷ ︸

µb0

+Uab

Uaaµa0(N)− 3

4Uabωa

Uaaωbhωa−Ubb(

√2sb)

−3

︸ ︷︷ ︸∆µb(N,0)

(4.21)

The third and fourth terms on the right hand side are the effective interaction between the main

BEC and the component B. The last term is the repulsion between the particles in component

B. The fourth term is small when ωb ωa and can be ignored. This result gives a very good

approximation as demonstrated in Fig. 4.4c.

Similarly, differentiating the chemical potential yields the second order term η2 in Eq. (4.15):

hη2(N) =12

25

µa0(N)

N︸ ︷︷ ︸∂Na µa(N,0)

−(

Uab

Uaaµa0(N)− 3

4Uabωa

Uaaωbhωa

)1N︸ ︷︷ ︸

∂Nb µa(N,0)

− 25

Uab

Uaa

µa0(N)

N︸ ︷︷ ︸∂Na µb(N,0)

+Ubb(√

2sb)−3

︸ ︷︷ ︸∂Nb µb(N,0)

(4.22)

The first three derivatives are smaller than the last term when ωb ωa and N→ ∞. Therefore, at

high ωb, the last term dominates η2(N), yielding

hη2(N)≈ Ubb

2(√

2sb)−3 =

Ubb

2

(mωb

2π h

)3/2. (4.23)

As shown in Fig. 4.4b, Eq. (4.23) gives an order of magnitude estimation of η2(N). Note that we

can also get the dominant term as calculated above by assuming component A to have a constant

density distribution |ψ(r)|2 = µa0/(NaUaa), at ωb ωa. A better approximation should take into

account the change in density ρa as shown in Fig. 4.5.

4.11 Appendix C: Effects of higher-order nonlinearities

Cat states can be distorted by higher order nonlinearities. Thus we need to find out to what extent

the cat states are distorted and whether the distortion is tolerable. Another practical problem is to

90

0.0 0.5 1.0 1.5 2.0 2.5 3.0

t/τc

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Qmax

CSS at t = 0

CAT at t = τ ∗c

CSS at t = 2τ ∗c

Figure 4.6: The maximum of the Q function, Qmax(β ), as a function of the relative time t/τc.The “best” real cat time τ∗c is defined as the time in which there are the two highest peaks inthe Q-function. The leftmost peak corresponds to the initial coherent spin state (CSS), with avalue Qmax = 1/π . The peak at τ∗c /τc ≈ 1.06 corresponds to the spin cat state (CAT). The time atwhich the CAT state is observed is shifted with respect to the ideal case τ∗c /τc = 1 because of thehigher-order nonlinearities. The highest peak at τ∗c /τc ≈ 2.12 corresponds to the CSS at the revivaltime. Note that all fitting orders are included for determining τ∗c . The CAT state at τ∗c and CSS at2τ∗c are plotted in Figs. 3a and 3b in the main text. The other parameters used are the same as inFig. 4.4.

figure out the optimal time to observe a cat state in real experiments. We define the “best” cat time

τ∗c as the time with the two highest peaks in Q function. This definition is based on the feature of

the cat states that two separated peaks in the Q function should be distinguished clearly.

This method is illustrated by Fig. 4.6 with the highest peak value Qmax plotted over time. It is

clear that the peak for the cat state is located near τ∗c /τc = 1 as expected. In practice, we search

around the nearby region, say τ∗c /τc ∈ [0.8,1.6], for the highest peak. The resulting τ∗c corresponds

to the best cat time. We further manually check that there are indeed only two opposite peaks in

phase space. The resulting shift in the cat time is plotted in Fig. 4.4d. Note that the τ∗c depends on

n, see also Fig. 4.4f.

In the scheme, the output light is of the form |χ(t)〉L = ∑n cne−iη(N,n)t |n〉L with the initial

91

−4 −2 0 2 4

Re(β)

−4

−2

0

2

4

Im(β)

0.000

0.033

0.066

0.099

0.130

Q(β)

(a) n = 9 and τ∗c = 1.005τc

−5 0 5

Re(β)

−5

0

5

Im(β)

0.000

0.031

0.063

0.094

0.130

Q(β)

(b) n = 49 and τ∗c = 1.026τc

−20 −10 0 10 20

Re(β)

−20

−10

0

10

20

Im(β)

0.0000

0.0084

0.0170

0.0250

0.0340

Q(β)

(c) n = 400 and τ∗c = 1.52τc

Figure 4.7: Plot of Q(β ,τ∗c ) for different cat sizes n. (a) n = 9. The third order nonlinearity η3 isweak, so the Q function looks like a perfect circle. (b) n = 49. The effects of η3 begin to appearand the cat state is distorted slightly. (c) n= 400. Both η3 and η4 are significant. The two peaks aredistorted and not symmetric. Note that τ∗c is not quite well defined in this case. The case n = 100is plotted in Fig. 3a in the main text with ωb = 2π500Hz and τc = 0.646s. The other parametersused are the same as in Fig. 4.4.

condition |χ(0)〉L = |α〉L and α =√

n. Hence, the Q-function without loss is

Q(s,θ , t) =1π

e−(α−s)2∣∣∣∣∑

n

((αs)n

n!e−αs

)e−inθ e−iη(N,n)t

∣∣∣∣2

(4.24)

where the phase space is defined by β = seiθ . This equation is numerically evaluated to obtain the

Q-function for given ηk, which are obtained by fitting the solutions of the GPE Eq. (4.6). A few

more figures corresponding to Fig. 3a in main text are plotted in Fig. 4.7 for different cat sizes n.

One can see that the higher order effects (k ≥ 3) are weak for small n, but significant for larger n.

4.12 Appendix D: Phase separated regime and non-phase separated regime

The scheme should also work in the non-phase separated regime a2ab < aaaabb. Fig. 4.8 shows

the coefficients η2, η3, η4 for different values of the inter-species scattering length aab, with

aaa = 2.8nm and abb = 3.4nm. The plots suggest that the Kerr effect is also strong in the non-

phase separated regime, but the higher order terms might limit the resulting cat size n. The main

qualitative difference is that there are no zero-crossing points for ηk in the non-phase separated

regime. These results further suggest that the weakly phase separated regime is advantageous

because the higher-order terms are very small around ωb/2π ≈ 400Hz.

92

50 100 150 200

ωb/2π(Hz)

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

η 2(H

z)

aab = 0

aab = 2.5nm

aab = 3.4nm

aab = 4.0nm

(a) η2(ωb) for different aab

200 400 600 800 1000

ωb/2π(Hz)

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

η 3(0.01H

z)

aab = 0

aab = 2.5nm

aab = 3.4nm

aab = 4.0nm

(b) η3(ωb) for different aab

200 400 600 800 1000

ωb/2π(Hz)

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

η 4(0.0001H

z)

aab = 0

aab = 2.5nm

aab = 3.4nm

aab = 4.0nm

(c) η4(ωb) for different aab

Figure 4.8: Effects of the cross-scattering length aab on (a) η2, (b) η3, (c) η4, with aaa = 2.8nmand abb = 3.4nm (aii is the self-scattering length of component i). Both aab = 0 and aab = 2.5nmare in the non-phase separated regime a2

ab < aaaabb, while aab = 3.4nm and aab = 4.0nm are inthe phase separated regime a2

ab > aaaabb. When aab is turned on gradually, the magnitude ofall nonlinear coefficients ηk decreases at first because the effective scattering length for the twocomponents decreases. All coefficients show a qualitative change, with a zero-crossing point in thephase separated regime. Compared with the non-phase separated regime, say, aab = 0, the phaseseparated regime can have a relatively weak higher-order effect even for high trapping frequencies,e.g. the small η3 at ωb/2π = 500Hz which is used in Fig. 3 of the main text. Note that the y axisis rescaled by factors of 100 from left to right for easy comparison. The parameters used are thesame as in Fig. 4.4, except aab.

4.13 Appendix E: Atom loss rates

The atom loss rate for component i is given by [69, 178, 179]:

dNi

dt=−τ

−1i,` =−

(L1,i

∫d3rρi + ∑

j=a,bL2,i j

∫d3rρiρ j + ∑

j=a,b∑

k=a,bL3,i jk

∫d3rρiρ jρk

)(4.25)

where L1,i, L2,i j, L3,i jk are the one, two and three particle collision loss rates. Note that the density

ρi = Ni|ψi|2 in the equation also depends on the numbers of particles Ni which decrease over time.

As discussed in the main text, the individual times to lose one particle through an m-body process

with particle combination c are defined as τ1,i = (L1,i∫

d3rρi)−1, τ2,i j = (L2,i j

∫d3rρiρ j))

−1 and

τ3,i jk = (L3,i jk∫

d3rρiρ jρk)−1, where L3,i = L1 and L3,i jk = L3 are used as an approximation. These

93

50 100 200 500 1000 2000 4000

ωb/2π(Hz)

0.001

0.01

0.1

1

10

τ(s)

τ3,baaτ3,bbaτ3,bbb

τ2,baτ2,bbτ1,b

τℓτc

Figure 4.9: Cat time τc and one-atom loss time τ` for Rubidium with n= 10, N = 105 and non-zerotwo-body loss rate L2,i j. It is clear that the time to lose one atom through the two-body loss withinthe same component, τ2,bb, dominates at high ωb, which has a similar scaling as the cat timeτc ∼ ω

−3/2b . This limits the maximum n to around 10 atoms. Parameters: Rubidium atoms 87Rb

with scattering length aaa = 100.44rB, abb = 95.47rB, aab = 88.28rB, where rB is the Bohr radius.The atom loss rates are L1 = 0.01/s, L2,aa = 0, L2,bb = 119×10−21m3/s, L2,ab = 78×10−21m3/s,L3 = 6×10−42m6/s [179], and ωa/2π = 20Hz.

time scales can be evaluated using numerical integration for the ground state density distribution

obtained from solving Eq. (4.6). Here, we further show the results for Rubidium atoms with non-

zero two-body loss rate in Fig. 4.9. The loss due to two-body effects is significantly larger than

that due to three-body effects in this case, which limits the maximum cat size to n = 10 atoms, as

compared to a few hundred atoms for the sodium example used in the main text. Note that this

is only one possible choice of states for Rubidium. Large cats may still be possible if appropriate

internal states and other conditions can be found such that two-body loss is suppressed.

For cat state creation, the maximum loss of atoms in the component B cannot be larger than

one atom. Therefore, we are trying to give a conservative estimation. Since |ψa(r)|2 ≤ |ψa0(0)|2 =

µa/(NaUaa) in the TFA, we can use the maximum |ψa0(0)| for the main BEC, and the Gaussian

94

50 100 200 500 1000 2000 4000

ω2/2π(Hz)

0.01

0.1

1

10

100

1000

τ3,baa

τ3,bba

τ3,bbb

τ3,baa,approx

τ3,bba,approx

τ3,bbb,approx

τ1,b

τℓ

τℓ,approx

τ(s)

Figure 4.10: Comparison of the numerical and approximation results for all loss processes corre-sponding to Fig. 2a in the main text. The solid curves show the numerical solutions of the time tolose one atom τm,c through an m-body process with particle combination c, while the approxima-tions are shown as dotted or dashed curves with the same color. The time to lose one atom throughall processes is calculated by τ` = (τ−1

1,b + τ−13,baa + τ

−13,bba + τ

−13,bbb)

−1. Note that the approximationshere are essentially lower bounds for the numerical solutions, as shown in this figure. See text fordetails.

φb0(r) for component B in Eq. (4.9):

∫ρbd3r = n (4.26)

∫ρ

2b d3r ≈

∫d3r|φb0|4n2 = (

√2sb)

−3n2 (4.27)∫

ρaρbd3r ≈(

µa0

Uaa

)n =

152/5π1/5

8N2/5

a3/5a s12/5

a

n (4.28)∫

ρ3b d3r ≈

∫d3r|φb0|6n3 = 3−3/2s−6

b n3 (4.29)∫

ρaρ2b d3r ≈

(µa0

Uaa

)∫d3r|φb0|4n2 =

152/5π1/5

16√

2N2/5

a3/5aa s12/5

a s3b

n2 (4.30)

∫ρ

2a ρbd3r ≈

(µa0

Uaa

)2

n =154/5π2/5

64N4/5

a6/5aa s24/5

a

n (4.31)

The estimations for three body loss are shown in Fig. 4.10, which suggests they are good lower

95

bounds for τ3,i jk and the time to lose one atom through all loss channels τ` = (τ−11,b + τ

−13,baa +

τ−13,bba + τ

−13,bbb)

−1. The estimation is better at small ωb, since the density of component A is not

repelled away so that the approximation |ψa(0)|2 ≈ |ψa0(0)|2 is good.

4.14 Appendix F: Readout loss

The readout loss from spin states to light is treated using the beam splitter model with a given

loss rate r2. The state passing through the beam splitter is |χout〉L = ∑nk=0 Bnk |n− k,k〉L with

Bnk = tn−krkn!/(k!(n− k)!), so the reduced density matrix ρ ′ is

ρ′ = Tr2(ρ) = ∑

i〈i|ψout〉L 〈ψout |i〉= ∑

n,m

min(m,n)

∑k

BnkB∗mkcn(t)c∗m(t) |n− k〉〈m− k| (4.32)

Hence, the resulting Q function with loss Qloss(s,θ , t) and initial coherent state |α〉L can be written

as:

Qloss(s,θ , t)=1π

e−(tα−s)2

∑m,n

(min(m,n)

∑k=0

(α2r2)k(tαs)n−k(tαs)m−k

k!(n− k)!(m− k)!e−(α

2r2+2tαs)

)e−i(n−m)θ e−i(η(n)−η(m))t

(4.33)

The term inside the big bracket is the bivariate Poisson distribution so this summation is upper

bounded by 1. Therefore the resulting Qloss(β ) is confined to the annulus |s− tα| ∼ 1. Hence, the

effect of photon loss is to move the peak of the Q-function toward the origin, as shown in Fig. 3c

and 3d in the main text. This form of the Q-function can be evaluated fairly efficiently with time

complexity of order O(n3/2), which allows us to evaluate it for cat sizes of order a few hundred

atoms.

4.15 Appendix G: Allowable uncertainty in atom number

Since all ηk(N) depend on the total atom number N, the statistical fluctuations in N can cause

dephasing (equivalent to angular spreading in phase space for the Q-function studied here), which

can wash out all observable features of cat states (consider for example the N-dependent rotation

96

0 200 400 600 800 1000 1200

ωb/2π(Hz)

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

η′ 1(N

)(Hz)

(a) dη1/dN vs ωb

0 200 400 600 800 1000 1200

ωb/2π(Hz)

−0.000012

−0.000010

−0.000008

−0.000006

−0.000004

−0.000002

0.000000

η′ 2(N

)(Hz)

(b) dη2/dN vs ωb

0 200 400 600 800 1000 1200

ωb/2π(Hz)

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

∆ϕ=

ϕ′ ∆

N

(c) Dephasing ∆ϕ with ∆N = 0.05N, at τc

0 200 400 600 800 1000 1200

ωb/2π(Hz)

0.0

0.1

0.2

0.3

0.4

0.5

∆N/N

=1/(N

ϕ′ )

(d) Allowable range of uncertainty in ∆N/N, at τc

Figure 4.11: Allowable range of atom number uncertainty with N = 105 and n = 100. (a) η ′1is basically constant over a large range of ωb. (b) η ′2 decreases with ωb. (c) Total dephasing∆ϕ = ϕ ′∆N, where ϕ ′ = (π/2η2)(∑k knk−1η ′k) includes up to fourth order terms from the GPE.∆ϕ needs to be smaller than 1 to have two distinguishable peaks of cat states, see Fig 3c and 3din main text for the Q-function for the case of ∆ϕ = 0.5 at ωb/2π = 500Hz. (d) The gray regionindicates the allowable uncertainty in atom number ∆N/N. It shows that the uncertainty in N canbe very large around ωb/2π = 600, and about 10% for high ωb. The other parameters used are thesame as in Fig. 4.4.

97

e−iη1(N)t caused by η1(N)). The dephasing is small if the derivatives of the coefficients with

respect to N, η ′k(N) = ∂Nηk(N), are small. These quantities are plotted in Fig. 4.11a and 4.11b.

Note that the constancy of η ′1 in Fig. (4.11a) can be understood from Eq. (4.21) because η ′1 =

(Uab/Uaa− 1)µ ′a0 is independent of ωb. Also, the dephasing is linear in time, hence, a short cat

time τc can significantly reduce the dephasing effects. Moreover the rotation generated by η1(N)

can be canceled by the opposite rotation generated by η2(N), as we derive below.

First considering the expansion of n = n+∆n around n the relevant terms become

η1(N)n+η2(N)n2 = (η1n+η2n2)+(η1 +2η2n)∆n+η2∆n2 (4.34)

On the right hand side, the first term gives a global phase which can be neglected. The second term

leads to a rotation in phase space. Writing N = N +∆N and expanding the coefficients around N

one has

ηk(N) = ηk(N)+η′k(N)∆N (4.35)

where η ′k(N) = ∂Nηk(N). Note that ∆ηk(N) = η ′k(N)∆N is the fluctuation in ηk due to the un-

certainty ∆N. Substituting these back into the second term in Eq. (4.34) yields the dephasing

term (η ′1∆N +2nη ′2∆N)∆n. This dephasing term is the source of a ∆N dependent rotation in the

β -plane, which is eliminated when the condition η ′1(N)+2nη ′2(N) = 0 is satisfied, see Fig. 4.11c.

In particular, we want to find out the maximum allowable ∆N that still preserves an observable

spin cat state at the cat time τc. Therefore, we define ∆ϕ = ϕ ′∆N = τc(η′1 + 2nη ′2)∆N, and the

condition |∆ϕ|. 1 should be satisfied, or

∆N .1ϕ ′

(4.36)

The higher order terms ηk can also be included, yielding

ϕ′ =

π

2η2(∑

kknk−1

η′k) (4.37)

Numerically, we find η ′k(N) by taking the numerical derivative of ηk(N). The results in Fig. 4.11d

show that there is a large range of allowable uncertainty in atom number ∆N if ωb is high enough.

98

−10 −5 0 5 10

−10

−5

0

5

10

0.000

0.014

0.027

0.041

0.054

(a) t = τc,L1τc = 0.01

−10 −5 0 5 10

−10

−5

0

5

10

0.000

0.011

0.022

0.033

0.045

(b) t = 2τc,L1τc = 0.01

−10 −5 0 5 10

−10

−5

0

5

10

0.0000

0.0044

0.0088

0.0130

0.0180

(c) t = τc,L1τc = 0.025

−10 −5 0 5 10

−10

−5

0

5

10

0.0000

0.0025

0.0051

0.0076

0.0100

(d) t = 2τc,L1τc = 0.025

Figure 4.12: Q function of continuous atom loss for the standard Kerr effect with HamiltonianH = hη2n2. (left column) At cat time t = τc = π/|2η2|, (right column) At revival time t = 2τc.Mean photon number n = 100 and 5000 samples.

Note that this range is an estimation since only the first order approximation of ∆ηk = η ′k(N)∆N is

used. In contrast, the accuracy requirement ∆N/N at low trapping ωb is even higher than the high

resolution of counting cold atoms of 1 in 1200 in a recent experiment [184].

4.16 Appendix H: Atom loss

The continuous loss of atoms from the BEC can be described by the master equation:

99

−10 −5 0 5 10

−10

−5

0

5

10

0.000

0.014

0.028

0.042

0.056

(a)

−10 −5 0 5 10

−10

−5

0

5

10

0.000

0.012

0.025

0.037

0.049

(b)

Figure 4.13: Q function of continuous atom loss for ηk calculated from GPE for the parametersgiven in Fig. 2b in the main text (n = 100, ωb = 2π500). (a) t = τ∗c = 1.06τc = 0.68s, L1 = 0.01/scorresponding to about 0.68 atom loss, (b) t = 2τ∗c , about 1.34 atom loss. No other effect isincluded. 5000 samples.

ρ =− ih[H ,ρ]+∑

i(RiρR†

i −12

R†i Riρ−

12

ρR†i Ri) (4.38)

where ρ is the density operator, H is the Hamiltonian of the system, R is the Lindblad operator

for the loss channel in question, and the summation is over the different loss channels. In the main

text, the 2-body loss is assumed to be zero and 3-body loss is much lower than the 1-body loss (see

Fig. 3) in the regime considered. Therefore, only 1-body loss will be considered below to simplify

the calculation. The loss atoms from the large component causes a change in N. The resulting

effects are similar to the fluctuations in N considered in the previous section. Here we therefore

focus on the small component. In this case one can effectively describe the system by the density

operator ρ = |χ(t)〉〈χ(t)|, with the initial state |χ(t = 0)〉 = |α〉 = ∑cn|n〉. Only one Lindblad

operator R =√

L1a is needed. The simplified master equation is:

ρ =− ih[H ,ρ]+L1(aρ a†− 1

2a†aρ− 1

2ρ a†a) (4.39)

with the Hamiltonian given by Eq. (4.8). The method used to simulate the system is the Quantum

Jump Method [185, 186]. The results are shown in Fig. 4.12. For the standard Kerr effect without

100

higher-order terms, it can be observed that the creation of spin cat state still results in two clear

peaks in the Q-function even when 2.5 atoms are lost on average. In fact, the cat state is still

visible even for an average loss of 5 atoms. However, for the detection scheme, the system is

required to evolve for 2τc, which limits the loss rate to L1τc < 0.025 as shown in the figure. For

the parameters used in Fig. 2b in the main text (including higher-order nonlinearities), the average

number of atoms lost is only 0.68 and the effect of the loss is small, see Fig. 4.13. The main effect

of the loss is a fairly uniform background ring in the Q function.

4.17 Appendix I: Comparison with photon-photon gate proposal

Ref. [160] utilizes a similar collision induced cross-Kerr nonlinearity in BEC to implement photon-

photon gates, while the current scheme uses a self-Kerr nonlinearity to create spin cat states. The

Kerr effect in the previous scheme is enhanced by increasing both scattering length (through a

Feshbach resonance) and the trapping frequency for both components. However, the Feshbach

resonance induced atom loss can be very large [69], which will limit the maximum cat size. Not

relying on a Feshbach resonance also makes it possible to use the magnetic field to further eliminate

atom loss. Also, both trapping frequencies should not be increased at the same time because it will

result in high atom loss through the collision with the main BEC. Instead, we suggest here to

increase only the trapping frequency of small component. This results in a similarly strong Kerr

effect, but with lower atom loss.

Moreover, the treatment in the previous paper, which used the quantized mean-field GPE with

TFA and first order perturbation theory, does not allow the study of higher order nonlinearities or

atom number fluctuations. Our present approach allows us to study both of these effects, and we

show that they can be significant. The assumption of equal trapping frequencies also limits the

previous treatment to the non-phase separated regime, which limits the choice of regimes with low

atom loss, such as the sodium atom example used here. Furthermore, the density of the stored

component in the previous scheme is much smaller (about four orders of magnitude) than the main

101

component. This raises the concern of other possible dominant effects on the same scale, such

as quantum depletion [176]. As we have shown, these problem can be minimized in the current

scheme by using a high enough ωb so that the small component is located at the center with high

density. Note that Eq. (4.23), which is obtained as a limiting case for high ωb here, is basically

equivalent to the results of the treatment in Ref. [160].

102

Chapter 5

Matter-wave mediated hopping in ultracold atoms: Chimera

patterns in conservative systems

5.1 Preface

Physical systems can often be well described by instantaneous nonlocal theories, such as gravita-

tional and Coulomb interactions, when the dynamics of the mediating field is orders of magnitude

faster. Based on the same principle, here, I propose a new mediating mechanism that can achieve

nonlocal spatial hopping for particles in systems with two inter-convertible states and very differ-

ent time scales. Adiabatically eliminating the fast component results in an effective hopping model

with independently adjustable nonlinearity, hopping strength and range. I show that the model can

be implemented in Bose-Einstein Condensates mediated by matter waves with current technology.

The results further show that the nonlocal hopping can result in non-trivial dynamical patterns

known as chimera states, characterized by coexisting regions of phase coherence and incoherence.

My analysis shows that chimera patterns can be observed in Bose-Einstein Condensates includ-

ing the mean-field limit, hence, presenting the first known evidence of conservative Hamiltonian

systems exhibiting chimera patterns.

This whole chapter contains my original work. I was attempting to find an analogue mecha-

nism of diffusive coupling in BECs, as well as the existence of chimera patterns in BECs. With

encouragement and guidance from both Prof. Simon and Prof. Davidsen, I was able to find the

answers to the question.

103

5.2 Introduction

Locality is one of the basic principles of physics, which constrains the finite speed propagation of

all perturbations and information. Despite this fact, a wide range of physical systems can be accu-

rately and conveniently described by the instantaneous nonlocal coupling such as magnetic dipole

interaction [187], Rydberg excitation [188], cavity-mediated coupling [71], and recent proposal for

tunable long-range interaction [189, 190] mediated by light. The picture of particle-field-particle

interaction can be reduced to an effective nonlocal particle-particle description when the mediating

field has a much faster time scale such that the state, motion, and separation of the particles can be

treated as constant.

Here, I introduce a new mediating mechanism to achieve the nonlocal spatial hopping, in

which particles can jump not only to its nearest neighbor sites, such as Bose-Hubbard model

[104, 190, 191], but much further away directly. This is possible if the particles can be converted

into a mediating channel experiencing no energy barrier between the neighboring sites as shown

Fig. 5.1a. Mathematically, the channel can be eliminated adiabatically, resulting in the nonlocal

hopping model (NLHM) with independently adjustable on-site interaction, hopping strength, and

hopping range. The mechanism is inspired by the diffusive coupling [6], but all physical process

are required to be coherent. As far as we know, the mechanism and realization are not yet known.

The new length scale of hopping radius, as we show, can result in a non-trivial dynamical

pattern with the phase coherent and incoherent region coexisting in the same state, generally known

as chimera states [5, 126]. This pattern can appear in laser arrays, chemical reactions, mechanical,

electronic, and neural networks. It is the result of the interplay between nonlinearity and nonlocal

couplings, with nonlocal Kuramoto model as the representative model. Only few chimera patterns

have been observed in experiments only recently [23, 21, 24], and they are less physical relevant.

Though the Hamiltonian of Kuramoto model is known recently [192], no Hamiltonian showing

chimera patterns are known. Here, we show three such Hamiltonians exhibiting chimera patterns.

In particular, we show the existence of chimera core pattern composing incoherent cores near

104

Figure 5.1: Illustration of nonlocal hopping. (a) Two-components model. Particles in traps areconfined so there are no spatial dynamics. Localized particles can be converted into the mediatingchannel that can propagate freely. It is eventually converted back to the nearby sites. (b) Effectivemodel. If the time scale of the mediating state is much faster than trapped state, the mediatingcomponent can be eliminated adiabatically and the hopping can be considered to be instantaneousand nonlocal with range R. (c) Periodic lattice with spacing d and depth V0. Trapped bosonicparticles can be described by the ground state wavefunction with width ` and energy ε1 (withenergy gap ε2− ε1). The hopping strength P and the hopping radius R can be controlled by theRabi frequency Ω and detuning ∆ = ∆2− ε1.

the spatial phase singularity in the coherent background [7, 18, 8], and properties unique to the

Hamiltonian system.

The physically realizable implementation we propose is based on the two components inter-

convertible Bose-Einstein Condensate (BEC) [9] in a spin-dependent trap [193], which is inspired

by Ref. [104] of the Bose-Hubbard model in cold atoms. In this setup, the atomic hopping process

is mediated by the matter wave itself in a way that the trapped component is isolated, while the

untrapped component is treated as the mediating channel, as illustrated in Fig. 5.1. This spatial

hopping is originated from the spreading in Schrodinger equation. Implementation in BECs has the

advantage of studying both quantum and classical regimes, and high controllability of almost all

105

parameters to a certain degree [85, 64]. For example, loss is important issue for general quantum

system [12] and limits the lifetime of system which turns out to limit the hopping range as we

show. It may be reduced by lowering the density and Feshbach resonance [69] which can change

both the nonlinear interaction and loss. We conclude that nonlocal hopping exists, and chimera

patterns may be observed within certain parameter regimes.

5.3 Nonlocal hopping

The Hamiltonian of the NLHM is

H = U +P =U2 ∑

i|ai|4−P∑

i, jGi ja∗i a j (5.1)

where ai =√

nieiθi is a complex number representing the state of site i with the number of particle

ni = |ai|2 and the phase θi. U is the nonlinear energy with the on-site nonlinear interaction U ,

and P is the hopping energy with the hopping strength P. Gi j is the hopping kernel describing the

hopping from site r j to ri, with Gi j =G ji and normalization ∑ j Gi j = 1. In typical physical systems,

Gi j decreases as the distance |r j− ri| increases and may be characteristic by a hopping range R.

For sufficiently small R, the hopping effectively becomes nearest neighbor. This Hamiltonian

can also be expressed in the canonical coordinate and momentum qi, pi, as well as action and

angle variable ni,θi (see Appendix). The later one should be more suitable for the study of phase

dynamics. Note that hopping is quadratic a∗i a j in the Hamiltonian which is different from the usual

quartic term of the particle-particle interaction nin j for, say, Coulomb interaction. Therefore, the

corresponding dynamical equation contains the lowest order on-site nonlinearity and the nonlocal

linear term:

ihai =U |ai|2ai−P∑j

Gi ja j (5.2)

The nearest-neighbor variation of this equation is the discrete Gross-Pitaevskii equation [194] and

the non-spatial variation is the discrete self-trapping equation [195].

106

Table 5.1: Hopping kernel GD(r) (to be normalized) with r = |r j− ri| in D dimension. K0 is themodified Bessel function of the second kind.

D 1 2 3GD(r) e−r/R K0(r/R) 1

r e−r/R

5.4 Mediating mechanism

The simplest model captured the concepts of mediating channel in Fig. 5.1a takes the following

form:

ihψ1(r, t) = U |ψ1|2ψ1 + hΩψ2 (5.3)

ihψ2(r, t) = −hκ∇2ψ2 + hΩψ1 + h∆ψ2 (5.4)

for the localized ψ1 and mediating ψ2 component respectively. Eq. (5.3) describes a localized

component with nonlinear interaction U and Rabi frequency Ω for Rabi oscillation, which is a

coherent conversion that conserves the particle numbers. Eq. (5.4) describes the mediating channel

with inverse mass κ = h/(2m) and detuning ∆ from the localized component. It is essentially

the Schrdinger equation with a coherent conversion, so the spatial propagation is originated from

the kinetic energy term. The additional detuning in the far-detuned regime |∆| |Ω| ensures

the mediating idea is well-defined: Number of particles Nk =∫

dr|ψk|2 in the mediating channel

N2 N1 ≈ N can be neglected.

Suppose ψ1 evolves much slower than ψ2, then the adiabatic elimination can be used by set-

ting ψ2 = 0 [196]. The solution of −κ∇2ψ2 +Ωψ1 +∆ψ2 = 0 in the unbounded isotropic space

with translation invariant [6] is given by the convolution ψ2(r, t) =−(Ω/∆)GD(r)∗ψ1(r, t) where

GD(r) is the hopping kernels listed in Table 5.1, with hopping radius R =√

κ/∆. Note that ∆ > 0

is required for the confined hopping kernels solution (see mediating state in Fig. 5.1a, while ∆ < 0

leads to a wave-like solution). Substituting this solution back to Eq. (5.3), we can get the contin-

uum NLHM where the summation is replaced by integral with hopping strength P = hΩ2/∆.

107

5.5 Implementation in ultracold atomic systems

The kinetic energy term in Eq. (5.3) cannot be ignored typically because coherent inter-convertible

particles usually have the same mass m. Instead, the effective mass can be increased by using a

periodic lattice. This can be done in ultracold atomic systems where the particles are forced to be

localized with a sufficiently deep trap, described by:

ihψ1(r, t) =(−hκ∇

2 +V1 +g11|ψ1|2)

ψ1 + hΩψ2 (5.5)

ihψ2(r, t) =(−hκ∇

2 + h∆2)

ψ2 + hΩψ1 (5.6)

for the localized ψ1 and mediating ψ2 component respectively, where gi j (i, j = 1,2) are the two

particles collision constant. Here, we assume g22 = g12 = 0. Vi is the spin-dependent potential

[193], with V1 periodic and V2 = 0. Even though it is similar to Eq. (5.18) with the extra kinetic

and potential energy term, solving it is not straightforward because adiabatic elimination fails to

work in the general form of Eq. (5.18). It is because the high energy level εi>1 is not evolving

slowly comparing with the mediating component, and large ∆ = ∆2− ε1 can resonate with high

energy level. This can be solved by confining the system to the local ground state ε1 and suitable

detuning ε2−ε1 ∆ |Ω| as shown in Fig. 5.11c. Under these constraints, we can show that Eq.

(5.18) and (5.19) reduce to the exact form of Eq. (5.2) with P = hΩ2/∆, hopping kernel GD(r) in

Table 5.1 and

R =CD

(d2`

)D2√

κ

∆(5.7)

where CD is a constant (see Appendix for the proof). Since the effective conversion region has a

characteristic length scale 2` in a lattice units with length d, so scaling with 2`/d is expected when

d 2`. Indeed, we have the effective scaling ∆→ ∆e f f = (2`/d)D∆. In this picture, ai is the

state variable of the localized wavepacket at site i. Hence, the kernel Gi j describes the matter-wave

mediated hopping by annihilating a wavepacket at site j and creating a wavepacket at site i.

By further assuming the trapping potential to be sinusoidal V1(x) =V0 ∑σ sin2(kxσ ) with wave-

length λ , wavenumber k = 2π/λ , lattice spacing d = λ/2 and trap depth V0. For sufficiently large

108

-1 0 1

Re(ai)

-1

0

1

Im(a

i)

−100 −50 0 50 100

x(y = 0)

0.0

0.2

0.4

0.6

0.8

1.0

〈|hi|〉

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2

〈θi〉

Figure 5.2: Chimera patterns in Hamiltonian system of NLHM. (a) Initial phase distribution withuniform density. (b) Phase θi(t = 100) over the 2D lattice. (c) Number of particle ni = |ai|2corresponds to (b). (d) Local phase space trajectory for x = −100 (blue) and x = −4 (red) at thecut y = 0. (e) Average hopping 〈|hi|〉 and average angular 〈θi〉 over time at the cut y=0. Hoppingstrength P/Un0 = 0.4 with hopping radius R = 16d after time Un0t = 100.

V0, the local ground states around trap minima can be approximated by a Gaussian φσ (xσ ) =

e−πx2/(2`2σ )/√`σ with `σ =

√π h/(mωσ ). In this setting, the nonlinearity is enhanced by the den-

sity as U = g11/Ve f f with effective volume Ve f f = 23/2`x`y`z, and numerical fitting gives CD ≈ 1

(see Appendix).

5.6 Dynamics and chimera patterns

Here, we focus on the regime with simultaneous weak hopping PUn0 and long hopping range

R d, where n0 is the average number of particles per site. This introduces a new length scale R to

system which may show different dynamics [7, 6]. Starting from the initial condition (IC) of spiral

phase as shown in Fig. 5.2a with uniform ni = n0 (see Appendix), the system evolves according

to Eq. (5.1) into a state with phase incoherent near the center but coherent far away as shown in

109

Fig. 5.2b and 5.2c (see Appendix). The same pattern also exist with different IC such as a vortex

with random phase near center (see Appendix). Also, when the hopping changes to the nearest

neighbor, this localized disturbance quickly spreads and interferes with each other (see Appendix).

It suggests the random core is a stable localized pattern. Moreover, the dynamics near the core are

significantly different from the coherent background which is clear in Fig. 5.2d of the local phase

space trajectories (more in Appendix). ai is regular at the point far from the core, but is irregular

near the core. Furthermore, the system is also scale invariant with a given R/L and the random core

scales linear as R [134] (see Appendix). These results are similar to what is known about chimera

cores, which exists in system with on-site nonlinearity for self-sustaining oscillators and diffusive

coupling through nonlocal linear term [4, 7, 6, 134, 5]. In contrast, the mean angular frequency

〈θi〉 of undamped oscillators here are not regular, see 5.2d. On the other hand, the hopping term

hi = ∑ j Gi ja j is expected to have hi ≈ 0 near the center because of the randomness. Moreover, hi

is smooth for the spiral initial condition.

As a conservative Hamiltonian system, NLHM describes a closed system with both energy and

the particle number N =∑i ni conserved, and display chimera properties. Also, the system has time

reversal symmetry. These lead to persistent fluctuations or ripples as observed in 5.2 which would

damp away in dissipative system quickly. In addition, the results of the backward time evolution

of the core region is very delicate. With a small perturbation, the background can evolve back to

nearly the same states at t = 0, but the core remain incoherent, which again signify the difference

between two regions (see Appendix) In a realistic system, nonlinear loss can appear in a system

which can be modeled by U →U− iUloss. As the simulation results suggest, chimera patterns can

still be observed even with Uloss/U = 0.02 at Un0t = 100.

5.7 Experimental settings

There are certain criteria that need to be satisfied by experiments in order to have a system de-

scribed by NLHM. To summarize, these include the far detuning regime ∆ Ω for the small

110

(a)1 102 104

∆

10−1

100

101

102

R/d

(b)0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.1

t(s)

0

500

1000

1500

2000

2500

N1 N1

N2

(c) (d)

(e) (f)

Figure 5.3: Chimera patterns in BEC. (a) Hopping radius R vs detuning ∆ for 87Rb with hyperfinestates |F = 1,mF =−1〉, |F = 1,mF = 0〉, optical lattice with s = 40 and λ = 790nm which givesd = 395nm and `x = `y = 0.22d. (b) Number of atoms in both components starting with an initialspiral (see Appendix). (c-f) State at t = 200ms. (c) Phase and (d) density of localized component.(e) Phase and (f) density of mediating component. Loss is not included in the simulation andthe lifetime is τ = 1.4s. Other parameters are ni(t = 0) = 10, ∆ = 2π × 64Hz, Ω = 2π × 16Hz,g11/h = 50µm3/s, `z = 50`x, with predicted Un0/h ≈ 2π × 24Hz, P = 2π × 16Hz and R ≈ 5d.No-flux condition.

111

number of particles in ψ2, and ∆ ε2− ε1 to prevent excitation. A good adiabatic elimination

needs a slow varying time scale for the localized component h∆Un0,P assuming all ni ∼ n0. In

addition, R > d for the hopping to be considered nonlocal. All of Ω,∆,U can be adjusted easily in

experiments, so does P. Also, theoretically, R ∼ ∆−1/2 can be arbitrary large by having particles

staying in the mediating channel for long time. Therefore, the actual upper bound of R is set by

the experimental lifetime τ and duration. Considering sufficiently deep optical dipole lattice with

s = 40 (expressing V0 = sER in recoil energy ER = hκk2) so that the direct hopping can be ignored

[104], the range of mediated hopping radius R for Rubidium is shown in Fig. 5.3a. Note that the

lower bound of R is determined by the energy gap ∆ ω = 2π×46kHz.

The regime with competitive P ∼Un0 is the most interesting. However, a BEC in 3D optical

lattice is naturally in the very strong nonlinearity regime Un0 P, which is U/h = 2π×2.23kHz

with g11 = 4π h2a11/m and natural s-wave scattering length a11. It can be reduced by the use of

both low density and Feshbach resonance, which can be adjusted by many orders of magnitude of

a11 in experiments [69]. The former method is preferable to be used before the latter one because

it can decrease both nonlinearity and two-particles collision loss at the same time. For a given

λ , decreasing density cannot be done in 3D optical lattices. However, in a lower dimension, the

non-lattice dimension (z-axis in our setting) can be weakly trapped to reduce the density, resulting

in a lattice of cigarette shape wavefunction [92, 197]. The dominant loss is the two-particle loss in

the localized component which gives an estimated Uloss = hL11/Ve f f and τloss = Ve f f /L11 with a

L11 two-particles loss rate (see Appendix) [12], so increasing `z can improve lifetime τloss ∼ `z in

2D.

The derivation of effective models implies that chimera patterns can also be observed in certain

parameter regimes for NLHM and the Eqs. (5.2) and (5.3-5.4) (see Appendix). The problem is if

such parameter regimes can be achieved experimentally. One such experimentally feasible regime

has been identified as shown in Fig. 5.3. Nonlocal hopping induced chimera patterns can be

observed, which are the results of full simulation of realistic Eqs. (5.18)-(5.19). With 10 atoms on

112

average, each site can be well described by amplitude and phase. The initial state can be prepared

from a uniform BEC, with V1 adiabatically turned on until the direct hopping is suppressed and

taken over by mediated hopping. A short light pulse induced energy shift can then be used to

create any desire initial phase, which is a spiral for Fig. 5.3. The system states and dynamics can

be detected by standard time of flight technique [78], matter wave interference [198] or optical

readout. As it is clear from simulations that random cores appear eventually. Similar patterns still

exist even when the experimental time is longer than the lifetime, suggesting that chimera patterns

should be observable in BEC.

5.8 Discussion and outlook

In brief, a new mediating mechanism of nonlocal spatial hopping is introduced, and the realization

details in BEC with the current technology is given. All calculations are based on classical Hamil-

tonians. Since all the physical mechanism chosen are coherent, and conserve energy and particles,

so the mediated hopping can happen in the quantum regime. Mathematically, all Hamiltonians

and dynamic equations can be quantized, and Eq. (5.1) becomes Bose-Hubbard model with medi-

ated hopping. At the single particle level, the loss problem is less important so the hopping range

can be even larger and even global hopping when system size is less than R. Chimera patterns in

open quantum system have been studied recently [199], and our results suggest the possibility that

chimera patterns may even exist in closed quantum systems, which requires a better understand-

ing of quantum synchronization [124, 200, 125]. Nonetheless, the correctness of our mean-field

prediction of chimera patterns in ultracold atoms can be justified by the existence of slight loss,

which causes the system to follow the classical trajectory [43]. The experimental technique can

generalize the Bose-Hubbard model with tunable hopping from nearest-neighbor to infinite-range,

which opens the door for the exploration of the new exotic condensed matter states similar to other

long range effects [201]. We hope that the work here motivates further studies on the nonlocal

hopping both experimentally and theoretically.

113

5.9 Appendix A: Hamiltonians

The Hamiltonian of the nonlocal hopping model (NLHM) is

H =U2 ∑

i|ai|4−P∑

i, jG jia∗i a j (5.8)

which can be represented in few different canonical variables under different transformations (see

[192, 202]). We can define the canonical coordinates and momenta to be qi and pi:

ai =1√2(qi + ipi) (5.9)

a∗i =1√2(qi− ipi) (5.10)

Hence, in the canonical coordinate and momentum qi, pi system:

H =U8 ∑

i

(q2

i + p2i)2− 1

2P∑

i, jGi, j(qiq j + pi p j) (5.11)

Similarly, we can define action ni and angle θi such that ai =√

nieiθi , or

ni =12(q2

i + p2i)

(5.12)

θi = tan−1 (pi/qi) (5.13)

Hence, in the action-angle ni,θi coordinate system:

H =U2 ∑

in2

i −P∑i, j

Gi, j√

nin j cos(θ j−θi) (5.14)

Note that action n` can be interpret as the number of particles. The conservation of the total number

of particles can be expressed as the constancy of the quantities ∑i |ai|2, ∑i(q2

i + p2i)

and ∑i ni. The

energy is conserved as there are no explicit time dependence. Lastly, the global phase is irrelevant

here, so the Hamiltonian is invariant under any global phase rotation ai→ aieiθ0 .

In the continuum limit, such as the result from the simplified two-components model in the

main text, the corresponding Hamiltonian can be obtained by replacing ai → ψ(r), ∑i →∫

dr,

114

∑i, j→∫ ∫

drdr′ and Gi, j→ G(r,r′). Explicitly, those Hamiltonians become:

H =U2

∫dr|ψ(r)|4−P

∫ ∫drdr′G(r,r′)ψ∗(r)ψ(r′) (5.15)

H =U8

∫dr(q(r)2 + p(r)2)2− 1

2P∫ ∫

drdr′G(r,r′)(q(r)q(r′)+ p(r)p(r′)) (5.16)

H =U2

∫drn(r)2−P

∫ ∫drdr′G(r,r′)

√n(r)n(r′)cos(θ(r′)−θ(r)) (5.17)

5.10 Appendix B: Ultracold atom with periodic lattice

Starting from the equations describing the ultracold atomic system:

ihψ1(r, t) =(−hκ∇

2 +V1 +g11|ψ1|2)

ψ1 + hΩψ2 (5.18)

ihψ2(r, t) =(−hκ∇

2 + h∆2)

ψ2 + hΩψ1 (5.19)

Note that the reference energy is arbitrary, but it is well known that the adiabatic elimination works

best when the first component evolves the slowest [196]. No such choice exists in the general form

of Eq. (5.18), but it exists when we confine the dynamic to the ground state of individual traps.

There are no simultaneously good basis for both equation, though, the good basis for the local-

ized and mediating equation are the Wannier basis and Fourier basis respectively. For the resonance

dynamics, it is easier to understand in the Wannier basis [104] which are the orthonormal basis

wmn(r) for the equation εmnwmn(r) =−hκ∇2wmn+V1wmn with periodic potential V1, where n is

the energy band index and m is the lattice site index. In this new basis, the wavefunctions are repre-

sented by ψ1(r, t) = ∑mn amn(t)wmn(r) and ψ2(r, t) = ∑mn bmn(t)wmn(r) respectively. Substituting

back to Eq. (5.18) and (5.19), we have

ihamn(t) = εmnamn +U |amn|2amn + hΩbmn (5.20)

ihbmn(t) = h∑kl

cmnklbkl + h∆2bmn + hΩamn (5.21)

115

where

εmn =∫

Vdr(hκ|∇wmn|2 +V1|wmn|2

)(5.22)

U = g11

∫

Vdr|wmn|4 (5.23)

cmnkl = κ

∫

Vdr∇w∗mn(r)∇wkl(r) (5.24)

In a periodic lattice V1, the eigenenergy εm1 = ε0 is a constant in the lowest band n = 1 if the trap

are sufficiently deep or there are no overlap between the Wannier function of the nearest neighbor

site. Hence, we can shift the energy ∆2→ ∆ := ∆2− ε0 using the transformation amn→ amne−iε0t .

If the energy gap is large εm2− εm1 ∆, then we can ignore the resonance with the higher band

index n > 1. Furthermore, suppose initially there are no excited states, i.e. amn(t = 0) = 0 for

n > 1, then no excited states will be populated because there are no resonance with those states,

written explicitly:

iham1(t) = U |am1|2am1 + hΩbm1 (5.25)

ihbm1(t) = h∑kl

cm1klbkl + h∆bm1 + hΩam1 (5.26)

ihbmn(t) = h∑kl

cmnklbkl for n > 1 (5.27)

In this form, all important dynamics are captured, and the localized component can be slow relative

to the mediating component.

5.11 Appendix C: Hopping Kernel

The adiabatic elimination of the mediating channel can be obtained by setting bmn = 0. Therefore,

we can find the hopping kernel by solving bm1 in the following equation

0 = ∑kl

cm1klbkl +Ωam1 +∆bm1 (5.28)

0 = ∑kl

cmnklbkl for n > 1

self-consistently by setting am1 = 1 at the center. This draw a direct analogue of finding the con-

tinuum hopping kernel GD(r) as described in the main text by setting ψ1(r) = δ (r). The effective

116

conversion region have length scale 2σ of the localized package, in each lattice unit with length

d. Therefore, it is expected that the only difference for the solution is with the effective scaling

∆→ ∆e f f = (2σ/d)D∆. So, the solution is bi1 =Ω

∆Gi j ∗a j1 and substituting back to first compo-

nent, we have the hopping strength

P = hΩ2

∆(5.29)

and Gi j takes the same form in the Table 1 in the main text with discrete normalization, and the

effective hopping radius as

R =CD

(d

2σ

)D2√

κ

∆(5.30)

where D is the dimension and CD is a constant.

The above result can be verified numerically. For the method to find to hopping kernel in an

optical lattice self-consistently, we solve the corresponding time dependent equation and find the

equilibrium solution. Hence, Eq. (5.28) with a time splitting method becomes

bm1(t) =− Ωam1−∆bm1 (5.31)

˙ψ2(q, t) =− hq2

2mψ2 (5.32)

for the conversion step and propagation step respectively. Both of them have exact solutions and

the basis change is preformed between each step. The hopping kernel Gi j can be found by setting

am1 = δm j, where j is the source lattice site (chosen to be the center of the lattice), and obtained the

equilibrium solution b∗i1, which gives Gi j = bi1. The Gaussian is used to approximate the lowest

band Wannier function as

wm1(r) = φ(r− rm)∼ e−|r−rm|2

2σ2 (5.33)

where σ defines a characteristic length of the Gaussian function. This describes the local ground

state that can be approximated by a harmonic oscillator, such as a deep sinusoidal trap. And the

transformation between the real space and the Wannier basis are given by

bm1 = 〈wm1(r)|ψ2(r)〉=∫

Vd3rφ(r)ψ2(r) (5.34)

117

(a)0 10 20 30 40 50

xm/d

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

b∗ m1

σ = 0.2

σ = 0.2, fitσ = 0.1

σ = 0.1, fitσ = 0.05

σ = 0.05, fit

(b)0 10 20 30 40 50

xm/d

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

b∗ m1

σ = 0.2

σ = 0.2, fitσ = 0.1

σ = 0.1, fitσ = 0.05

σ = 0.05, fit

(c)0 1 2 3 4 5 6√

1/σ

0

1

2

3

4

5

6

7

8

R

(d)0 2 4 6 8 10 12 14 16 18

1/σ

0

2

4

6

8

10

12

R

Figure 5.4: Discrete hopping kernel Gi j. (a) Comparing b∗m1 ∼ Gm j with exponential fitting in1D. κ = 100, Ω = 10, ∆ = 10. (b) Comparing b∗m1 with K0 fitting in 2D. (c) The scaling of the ofthe hopping radius R∼ σ−1/2 in 1D vs the width of the Gaussian σ . (d) The scaling of the of thehopping radius R∼ σ−1 in 2D vs the width of the Gaussian σ .

where V is the confined volume around the lattice minimum [−d/2,d/2]D with a finite cutoff of

lattice spacing d.

The numerical results fit perfectly for the kernel GD(r) as shown in Fig. 5.4. With sufficiently

narrow Gaussian σ d, the predicted hopping radius R fits perfectly with Eq. 5.30.

5.12 Appendix D: Chimera patterns

The dynamic equation of NLHM can be rewritten in the dimensionless form using the rescaling

ai = ai/√

n0, t = (Un0/h)t, and P = P/(Un0) where n0 is the average particle number per site.

The equation becomes, after dropping the tilde,

i∂tai(t) = |ai|2ai−P∑j

Gi ja j (5.35)

which depends on the control parameters of rescaled hopping strength P and rescaled hopping

radius R = R/d. Without the hopping term, the system is decoupled and evolves locally as ai(t) =

118

e−i|ai(0)|2tai(0). The dynamics may be clear if the rotation is eliminated by shifting the reference

energy as:

i∂tai(t) =−ai + |ai|2ai−P∑j

Gi ja j (5.36)

which gives a solution ai(t) = e−i(|ai(0)|2−1)tai(0) with the extra global phase eit when P = 0. If the

system is uniformly distributed |ai(0)|2 = 1, then ai(t)= ai(0). Hence, the oscillation dynamics can

be eliminated in this reference frame, which is used in all simulation for NLHM. The simulation

is done in the square lattice with size L with no-flux boundary condition. The numerical method

used is the 4-th order Runge-Kutta method.

For the spiral initial condition, uniform density ai =√

n0 is used and the state is given by

ai(t = 0) =√

n0ei(ksr−tan−1(y/x)) (5.37)

with r =√

x2 + y2 and spiral wavelength ks. The dynamics is shown in Fig. 5.5. As shown in the

figure, the chimera core formed near the spatial phase singularity. This pattern is similar as the

system scaled up with R/L fixed as shown in Fig. 5.7 with 4 times larger system. The dynamics

outside the core is the same as clear indicated by the ripple near the core, but the core now becomes

4 times larger with random phase. This chimera core pattern can exist with alternative initial

condition such as a vortex with a randomized core of size radius Rrc as shown in Fig. 5.8. Note

that with large ks, a new pattern of chimera rings in 2D can be formed that surrounds the center

core as shown in Fig. 5.10.

The chimera core pattern depends on the nonlocal hopping range R d, whose dynamics are

very different from R ∼ d. The difference is very clear when the system starts from the random

core vortex with nearest neighbor hopping as shown in Fig. 5.9. The random phase near the core

is a highly localized disturbance that are eventually propagating outward and interferes with each

other. No localized chimera core pattern appears and the dynamic is different.

The inverse time propagation can go back to the initial condition by reversing the parameters.

Using the state in Fig. 5.5f as the initial condition, the system is propagated by the same amount of

the time t. As shown in 5.11a, it can perfectly go back to a spiral as expected. The chaotic nature

119

of the core can be shown by adding a single shot phase noise to the system state in Fig. (5.5)f as a

perturbation before the inverting time propagation occurs, given by

θi→ θi +χnoiseξi (5.38)

where the noise is Gaussian with 〈ξi〉= 0, 〈Re(ξi)Re(ξi′)〉= δi,i′ , and 〈Im(ξi)Im(ξi′)〉= δi,i′ , with

amplitude χnoise. As shown in Fig. 5.11b and 5.11c, the system cannot go back to the spiral

with noise as low as χnoise = 10−11. This suggest that the core regions are very sensitive to the

initial condition. This is in stark contrast with the coherent background which has almost the same

values as the noiseless case. It indicates that the system behaves differently in the core region and

the non-core which is the important property of chimera states.

As shown above, the minimal two-component model can be reduced to the effective NLHM.

The accuracy of the parameters mapping is tested and shown in Figs. 5.12 and 5.13 for 1D and

2D system. The results are already very good for the short time scale. The accuracy can be further

increases by increasing the detuning ∆, and decreasing the spacing dx.

5.13 Appendix E: Numerical methods

For the full simulation of the BEC in an optical lattice, Eq. (5.18) and (5.19), we use the time-

split spectral method for the two components Gross-Pitaevskii equation [203], with a 4-th order

time splitting scheme. For the simulation of the effective NLHM, Eq. (5.2), we use the 4-th order

Runge-Kutta scheme with a convolution in the Fourier space.

5.13.1 Split method

A two step time split method for the differential equation of the form

ψ = (A+ B)ψ, (5.39)

with operator A and B is given by

ψ(r, t +∆t) = ebs∆tBeas∆tA...eb1∆tBea1∆tAψ(r, t), (5.40)

120

Figure 5.5: Time evolution of initial spiral with spiral wavelength ksd = 0.01 with lengthL = 256d. Nonlocal hopping P/(Un0) = 0.4 and R = 16d in L = 256d.

121

Re(ai)−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

Im(a

i)

x = −128

Re(ai)

Im(a

i)

x = −15

Re(ai)

Im(a

i)

x = −13

Re(ai)−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

Im(a

i)

x = −10

Re(ai)

Im(a

i)

x = −6

Re(ai)

Im(a

i)

x = −5

Re(ai)−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

Im(a

i)

x = −4

Re(ai)

Im(a

i)

x = −3

Re(ai)

Im(a

i)

x = −2

Re(ai)−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

Im(a

i)

x = −1

Re(ai)

Im(a

i)

x = 0

Re(ai)

Im(a

i)

x = 1

−1.5−1.0−0.5 0.0 0.5 1.0 1.5

Re(ai)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

Im(a

i)

x = 2

−1.5−1.0−0.5 0.0 0.5 1.0 1.5

Re(ai)

Im(a

i)

x = 4

−1.5−1.0−0.5 0.0 0.5 1.0 1.5

Re(ai)

Im(a

i)

x = 8

Figure 5.6: The local trajectory of ai for Fig. 5.5 at different sites at y = 0. Time between t = 0and t = 100.

122

Figure 5.7: Similar to Fig. 5.5 in a larger system P/(Un0) = 0.4 and R = 64d in L = 1024d att = 100.

Figure 5.8: Time evolution of initial vortex with random core Rrc = 16d with nonlocal hoppingP/(Un0) = 0.4 and R = 16d in L = 256d.

123

Figure 5.9: Time evolution of initial vortex with random core Rrc = 8 with nearest neighborhopping, P/(Un0) = 0.4.

124

(a) t = 0 (b) t = 100

(c) t = 220 (d) t = 4000

Figure 5.10: Time evolution of NLHM with a spiral IC and wavenumber ks = 0.05. P/(Un0)= 0.1and R = 40d in L = 400d . BC: no-flux, kernel: top-hat.

125

Figure 5.11: Inverse the time simulation with initial condition given by Fig. 5.5f. (a) No noise,(b) χnoise = 10−11, (c) χnoise = 10−10, where a single shot noise is add at t = 0.

Table 5.2: Time splitting: First order coefficientsstep i ai bi

1 1 1

where ai, bi are coefficients satisfied ∑i ai = ∑i bi = 1. The eas∆tA and ebs∆tB in the equation are the

formal solution to the differential equations

ψ = Aψ, (5.41)

ψ = Bψ, (5.42)

respectively. For the coefficients of the first order, the second order Strang splitting, and a fourth

order method, see the Table 5.2, 5.3 and 5.4. Note that the time split method with negative coeffi-

cients, such as the one in Table 5.4, requires the differential equation to be time reversed.

Table 5.3: Time splitting: Second order Strang splitting coefficientsstep i ai bi

1 0.5 12 0.5 0

126

−100 −50 0 50 100

x

0.00

0.05

0.10

0.15

0.20

0.25

0.30

ρ1(x, t = 0.0)ρ2(x, t = 0.0)ρr(x, t = 0.0)

(a) t = 0

−100 −50 0 50 100

x

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

ρ1(x, t = 25.0)ρ2(x, t = 25.0)ρr(x, t = 25.0)

(b) t = 25

−100 −50 0 50 100

x

0.00

0.05

0.10

0.15

0.20

0.25

ρ1(x, t = 50.0)ρ2(x, t = 50.0)ρr(x, t = 50.0)

(c) t = 50

−100 −50 0 50 100

x

0.00

0.05

0.10

0.15

0.20

ρ1(x, t = 100.0)ρ2(x, t = 100.0)ρr(x, t = 100.0)

(d) t = 100

Figure 5.12: Comparison of NLHM and the minimal two-component model in 1D. Time evolutionof a initial Gaussian state with wide 20 for the minimal model (ρ1 and ρ2) and the NLHM (ρr).Common parameter for both model: Natom = 10, U = 1 with spatial discretization x = [−100,100],dx = 1. For minimal model, κ = 4000, Ω = 1, ∆ = 10, resulting in effective hopping rangeR0 =

√κ2/∆ = 20, P = Ω2/∆ = 0.1.

Table 5.4: Time splitting: Fourth order coefficients (Emb 4/3 BM PRK/A [204])step i ai bi

1 0.0792036964311954608 0.2095151066133618912 0.353172906049773948 -0.1438517731798180773 -0.0420650803577191948 0.4343366665664561864 0.219376955753499572 0.4343366665664561865 -0.0420650803577191948 -0.1438517731798180776 0.353172906049773948 0.2095151066133618917 0.0792036964311954608 0

127

(a) NLHM (b) Minimal model

Figure 5.13: Comparison of NLHM and the minimal two-component model in 2D. The phasesat t = 100 are shown with the same initial spiral ks = 0.01. Parameters for NLHM are Un0 = 1,P = 0.5, R = 16, and size L = 256 with no-flux boundary condition. Parameters for minimal modelare ∆ = 8, Ω = 2, U = 1, and κ = 2048, with dx = 1.

5.13.2 Time split method for the one-component GPE

A one component GPE is given by

ihψ(r, t) =− h2

2m∇

2ψ +V (r)ψ +U |ψ|2ψ. (5.43)

The splitting used is chosen to be

ihψ(r, t) = − h2

2m∇

2ψ, (5.44)

ihψ(r, t) = V (r)ψ +U |ψ|2ψ. (5.45)

The second equation is a pure local phase evolution without change in the local amplitude |ψ(r, t)|

or, similarly, the local density |ψ(r, t)|2, because

ihρ(r, t) = ih(ψ∗ψ +ψ∗ψ) = 0. (5.46)

Hence, the exact solution at the subsequent time t starting from t = 0 is given by

ψ(r, t) = ψ(r,0)e−i(V (r)+U |ψ(r,0)|2)t/h. (5.47)

128

The first equation can also be solved exactly, by transforming it in the Fourier space:

ih ˙ψ(q, t) =h2q2

2mψ, (5.48)

which has an exact solution

ψ(q, t) = e−i hq22m t

ψ(q,0). (5.49)

The real space solution can be obtained by the inverse Fourier transform.

5.13.3 Time split method for the two-component GPE

The two-component GPE considered is

ihψi(r, t) =−h2

2mi∇

2ψi +Viψi +∑

jgi j|ψ j|2ψi + hΩψ3−i + h∆iψi, (5.50)

with i = 1,2. Here, we only consider the equation without cross nonlinearity g12 = 0, as it is easier

to have an exact solution in the time split method. Otherwise, it can be solved by a three-step

time-split method in this case, or one can use a higher order numerical approximation for each

individual step. The following splits are considered

ihψ1 = − h2

2m1∇

2ψ1 + hΩψ2, (5.51)

ihψ2 = − h2

2m2∇

2ψ2 + hΩψ1, (5.52)

ihψ1 = V1(r)ψ1 +g11|ψ1|2ψ1 + h∆1ψ1, (5.53)

ihψ2 = V2(r)ψ2 +g22|ψ2|2ψ2 + h∆2ψ2, (5.54)

Similarly to the one component equation, the exact solutions to the second set of equations are

ψi(r, t) = e−i(Vi(r)+gii|ψi(r,0)|2+h∆i)t/hψi(r,0). (5.55)

The first equation in the Fourier space takes the form

i ˙Ψ(q, t) = MΨ, (5.56)

129

where M is a symmetric matrix given by

M =

hq2

2m1Ω

Ωhq2

2m2

Ψ(q, t) =

ψ1(q, t)

ψ2(q, t)

, (5.57)

which can be diagonalized. Hence the solution is

Ψ(q, t) = e−iMtΨ(q,0). (5.58)

These solutions can then be used in the time split method.

130

Chapter 6

Conclusion and future work

6.1 Conclusion

In this thesis, I have presented my studies on three nonlinear effects and attempted to propose

implementations using ultracold atoms. After studying chimera knot states in 3D and cat states in

Bose-Einstein Condensates, I intended to find an analogue for diffusive coupling in BECs. This

led to my discovery of a new matter-wave mediated nonlocal hopping mechanism.

In chapter 3, I have presented strong numerical evidence that stable knots and links can exist in

oscillatory systems and systems with nonlocal coupling. These include structures such as trefoils,

Hopf links, and more. The same conclusion holds in simple, complex and chaotic oscillatory

systems, if the coupling between the oscillators is neither too short-ranged nor too long-ranged. In

the case of complex oscillatory systems, I have also discerned a novel topological superstructure

combining knotted filaments and synchronization defect sheets.

In chapter 4, I proposed a method to create spin cat states, which are macroscopic superposi-

tions of coherent spin states, in BECs, using the Kerr nonlinearity due to atomic collisions. This

proposal includes an enhancement of the nonlinearity through the strong trapping of the small BEC

component, and an enhancement of the lifetime using a Feshbach resonance. Based on a detailed

study of atom loss, I concluded that cat sizes of hundreds of atoms should be realistic. The de-

tailed analysis I presented also includes the effects of higher-order nonlinearities, atom number

fluctuations, and limited readout efficiency.

In chapter 5, I proposed a new mediating mechanism that can achieve nonlocal spatial hopping

for particles in systems with two inter-convertible states and very different time scales. Adiabat-

ically eliminating the fast component results in an effective hopping model with independently

adjustable nonlinearity, hopping strength, and hopping range. I showed that this model can be im-

131

plemented in BECs using current technology. My results further show that the nonlocal hopping

can lead to chimera states, which should be observable in BECs experimentally.

6.2 Future work

My results in Chapter 3 suggest that the nonlocal diffusive coupling can stabilize the knot structures

in oscillatory media, and, in Chapter 5, nonlocal hopping is an analogue to the nonlocal diffusive

coupling. Technically, the Gross-Pitaevskii equation can be considered as describing a special case

of oscillatory media with energy and particle conservation, so it is natural to speculate that chimera

knots can also exist in systems with nonlocal hopping. The simulations, however, will take much

longer, because of high accuracy requirements for conservative systems, and will require detailed

planning on how to use the available computational resources as efficiently as possible.

For the spin cat states in BEC studied in Chapter 4, it will be interesting to really see it imple-

mented in an experimental system. Since the paper was published, we have been in contact with

experimental groups to see if it is feasible with their experimental setups. We have learned that

not many groups have exactly the same setting used in my paper, but that there should be no real

obstacle from the experimental point of view. However, research groups will need time to adapt

their experimental systems.

The study of mediated nonlocal hopping and the proposed implementation in BEC brings up

many questions. There are two main future directions for the mediated hopping. The first direction

is the search for alternative experimental implementations. The scheme proposed in Chapter 5 is

only one of the promising implementations found during my research. For example, an alternative

proposal might replace hyperfine states with a double well potential, which has the same mathe-

matical description. However, it requires one spatial dimension for the double well control, so it

will be harder to control loss. Another example might be a adiabatically and periodically varying

trapping, which will likely require a longer experimental time, and whose side effects are harder to

analyze. There are two other promising setups that I would like to study. My preliminary analysis

132

suggests that photon mediated nonlocal hopping with BEC in multiple traps should be possible.

This setup should have a faster time scale with no requirement of spin dependent traps, but it relies

on recently developing trapping technology [189, 190]. Another approach is a purely photonic sys-

tem using an optical-fiber based implementation, this is only possible if the nonlinearity is strong

enough compared with the loss.

Another direction is the study of the nonlocal hopping model itself. It is interesting to study the

dynamics of chimera states in conservative systems. Preliminary results show that there are more

spatial-temporal chimera patterns in BECs that have not been reported elsewhere, such as stable

chimera rings in 2D. There may be even more patterns in other parameter regimes that are yet to

be explored. The existence of chimera states in conservative systems also suggests a relation with

chaotic dynamics in conservative Hamiltonian systems [205, 206] confined to a spatial region.

The incoherent region shows some properties of superfluid turbulence. Further simulations also

suggest the existence of certain localization effects in systems with the nonlocal hopping. Further

work will be required to fully quantify these observations.

The effective model used in my analysis is the mean-field of the Bose-Hubbard model [13]

with nonlocal hopping. This raises questions related to condensed matter physics. In the standard

Bose-Hubbard model with strong hopping, the system is in the superfluid state with particles freely

moving, and is described by a macroscopic wavefunction. My scheme further constrains the wave-

function to exhibit periodically localized structures. This order is generally known as the lattice

supersolid state [191, 207, 208]. The current system in Chapter 5 is in the mean-field limit, so a

quantum treatment will be needed to quantify and prove the existence of the supersolid state.

For conservative Hamiltonian systems, eigenstates, and in particular the ground state, are well-

defined. This is generally not the case for chimera states in open systems. For the eigenstates, the

time evolution involves a time-dependent global phase with a spatial structure ψ(x). So it will be

interesting to study if such pure spatial chimera patterns can exist too, in addition to the typical

chimera states of spatio-temporal patterns.

133

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